It's a good question and one stimulating to my interest in comprehensive and holistic treatments of Islamic doctrine based on philosophical insights. So prepare yourself for a substantial intellectual ride! But before setting off, here's the short answer:
In Islam, hijab has always been obligatory. From an intellectual
perspective, full, integral modesty is not possible without hijab
because, in principle, virtues such as modesty and chastity have
external concomitants that have to observed, abandoning which leads to
moral ills that are contrary to that virtue.
From an Islamic historical and Quranic perspective
First of all there's no question that Islam enjoins Muslim women to wear hijab in order to uphold modesty and chastity. There are several Quranic verses and hadiths to this effect (See verses 24:31 and 33:59; and this and this hadiths). Consequently, all prophet's wives and early Muslimas and all traditional Muslim women to this day have observed the hijab and have seen it as a religious obligation. So with this Quranic and historical precedence, one would have a very difficult time arguing that hijab is not an Islamic obligation and concomitant for modesty and chastity. One can say he or she simply doesn't like hijab even though it's part of Islamic code of conduct, but one can't deny that it's an Islamic doctrine.
This would be an answer from an Islamic historical and literal perspective. But if one is interested in examining the relation between hijab and modesty and chastity rationally, here's an extensive discussion.
Some terminology
First, I'd rather start with some terminology. Based on my sources, "modesty" is a poor translation for the word حیاء (transliter. haya) in Arabic which means: to restrain one's self from doing a wrong out of a fear of reprehension and blame. So it implies a sense of moral shame. The opposite of haya in Arabic is وقاحة (transliter. waqaha) which means shamelessness. In Islamic context, haya would translate to a sense of restraint against something that according to Islam is reprehensible and shameful.
However "modesty" in prevalent usage can also be translated to or confused with عفة (transliter. 'iffa) that usually translates to "chastity". عفة literally means "remnant" and in basic idiomatic usage, it meant: to suffice oneself to a little sustenance that is left. In ethical context, it means preserving one's desires from excessive or inappropriate operation.
These were the general meanings of the words. But people and ethical schools can differ on practical implications and particular requirements of chastity and modesty, and even on whether chastity and modesty are good things at all.
An Islamic philosophical discussion
In several philosophies and traditions, there's a principle of moral moderation that is called The Golden Mean rule which indicates a desirable middle between two extreme levels of emotion, one of excess and one of deficiency (to go by Wikipedia example, think of courage positioned between recklessness and cowardice as two extremes). Morality is thus defined as moderating human desires towards the middle state. But while moral philosophers generally agree with this broad principle of moderation, they can differ widely on the actual requirements of moderation of each emotion.
But Muslim philosophers of ethics believe that only Allah knows the full requirements of the general ethical principles such as chastity that could nonetheless be comprehended and elaborated by human beings through deep contemplation. These scholars usually drew upon the revealed doctrine, the general framework of the Golden Mean Ethics and the Islamic discipline of knowledge of nafs that is usually associated with Sufi works to explain the nuances of Islamic ethics. Next would be an intellectual explanation of the Islamic requirements of modesty and chastity.
Contemplating the wisdom of hijab
Men and women have sexual desires each operating in distinct ways. Women's sexual desire influences them to attract men to different aspects of their natural beauty, (whether their body or their look or their voice or gestures), and that's how they find sexual satisfaction. Men's sexuality on the other hand inclines them to draw sexual pleasure from the attraction of women.
Hence on the operational level, women primarily lure and men primarily covet under the influence of their respective sexuality, although this mechanism can also operate conversely i.e. men trying to lure women and women getting attracted to men, but it is usually subordinate to the former pattern of cross-sexual interaction.
But since human sexuality is meant to fulfill some noble, vital purposes (under the light of the greater wisdom of Divine creation) such as the benefits of mutual life, reproduction, upbringing of children etc, Islam enjoins women and men to confine all their sexual desires and emotions — and generally everything that can be influenced by one's sexuality — to their married life. In other words, according to the Divine wisdom of human nature and purpose, sexuality is not an end in itself but it's a means to an end, and therefore this desire as a means has to be restricted and moderated so as to serve the specified end. So operating our sexuality in any other area, would be a deviation from the wise purpose decreed by the Creator for it.
Now, given the opposite operation of sexuality in women and men and in line with the sacred purpose of human sexuality, women should avoid anything that exposes them to sexual desires of anyone other than their married husband, and men should avoid anything that sexually stimulates them towards women other than their wives. "Anything" in the Islamic maximalist moderation can include absolutely anything, from revealing any amount of one's body that can arouse sexual attention such as hair, arms and legs, to more private body parts, to coquetry in gestures and speech.
This Maximalism view of chastity has traditionally also required women to cover their faces and hands if they thought they were likely to attract men sexually, hence pious women throughout history including Prophet's wives and female relatives preferred to cover their faces and hands when going outside, i.e. literally covering themselves from head to toe.
The dictates of our present civilizational epoch and a Messianic sidenote
Before proceeding to the requirements of Islamic chastity for men, I want to explicitly acknowledge that today many Muslims and non-Muslims find observing this maximum level of modesty and even the more miminal levels (such as sufficing to headscarves) to be excessive especially those raised in more secular or liberal environments, so here are some additional points to consider.
One way of looking more charitably at hijab is that when practiced out of an intellectual and heartfelt devotion, it can be regarded as an act of utmost moral sacrifice by a woman who embraces it with the intention of eliminating any ground for potential unchaste behavior in society.
Another point that helps to digest this position is that what looks to be an excessive or extreme form of chastity under the circumstances of this day and age, used to be completely normal in past traditional society for centuries. The change of habit and perception that many Muslims have experienced on this issue amongst many others over the last century has been due to the increasing secularization of Muslim Ummah under the secularization of modern civilization. Now since there are endless indications that this civilization is leading the entire humanity into abyss and is destined to collapse sooner or later anyway, it's upon us to decide whether we want to follow the norms of a doomed civilization or want to return to our traditional norms that are based on the Divine infallible wisdom of human nature and well-being.
Complying with the increasing diversion of the mainstream civilization from religion is to partake in the dark destiny that is awaiting it, but struggling to preserve our Islamic traditional way of life means breaking away from this ominous universal trend, and aligning ourselves with the forthcoming Divine civilization that is going to replace the current abysmal destiny as foretold by Mahdi traditions in Islam.
Finishing off the philosophical contemplation on hijab and Chastity
Now going back to my discussion of requirements of chastity. One-sided emphasis on women's modesty is always prone to charges of discrimination and sexism. So to prevent that misconception, and in line with the understanding of the distinct ways that each gender's sexuality operates, it has to be said that man's part in preserving sexual ethics is through moderating their own natural sexual inclinations, i.e. avoiding lustful looks at women whether done blatantly or sneakily. Women should not forget that just as it is difficult for them to moderate their natural inclination to attract men by revealing their beauties, it is equally difficult for men to constantly guard their eyes and desires in public, it only happens that for women the requirement is more physically "burdensome" since they need to cover themselves but on the upside women are generally more patient and adaptive than men! So non-observing women can very soon get accustomed to covering themselves with hijab with just as ease and convenience of practicing women.
One last more general point that I want to add, is that under the light of Islamic spiritual psychology, moderation of one's desires leads to intellectual strength and will power, since it is by the power of intellect that desires are moderated and it is as a result of the excess in desires that intellect and will power weaken. This is the reason for the general observation that emotionally principled people are wiser and more intelligent.
I hope seen within the above more "holistic" view of chastity within the greater scheme of Islamic wisdom, your friend can come to appreciate the gift of hijab and more fundamentally Islamic observance in general!
