Islam discourages unnecessary mixed-gender intermixing:
No person (man) should be alone with a woman except when there is a Mahram with her.
Sahih Muslim 1341 c
A man is not alone with a woman but the third of them is Ash-Shaitan.
Jami` at-Tirmidhi 2165 [grade: sahih]
So relationships like "girlfriend" and "boyfriend" are not appropriate, and would be considered haram (forbidden):
In Islam, the male-female relationship is based on chastity. It is allowed to communicate with the opposite sex if the manners prescribed by Islamic law (shari'ah) are observed and for legally allowed purposes such as study, work or social activities. If a relationship exceeds these restrictions, it becomes prohibited.
Dar Al-Ifta
There's a zillion fatawa on topics like this, and they almost uniformly declare such relationships either haram or as temptation (leading to haram).
In practice, however, there are many mixed-gender interactions taking place between Muslims and non-Muslims. Some Muslims can be a bit relaxed about the rules, and trust in Allah's forgiveness for minor inappropriate mixed-gender interactions (as per Qur'an 4:31). But after this, there's some hard barriers.
The Qur'an directly forbids marriage to a polytheist (and Hinduism has multiple deities):
And do not marry polytheistic women until they believe. And a believing slave woman is better than a polytheist, even though she might please you. And do not marry polytheistic men [to your women] until they believe. And a believing slave is better than a polytheist, even though he might please you. Those invite [you] to the Fire, but Allah invites to Paradise and to forgiveness, by His permission. And He makes clear His verses to the people that perhaps they may remember.
Qur'an 2:221
So marriage is clearly haram (unless the man converts to Islam).
Such a marriage would not be Islamically recognized, and sex would be considered fornication for which there is a prescribed punishment:
The [unmarried] woman or [unmarried] man found guilty of sexual intercourse - lash each one of them with a hundred lashes, and do not be taken by pity for them in the religion of Allah, if you should believe in Allah and the Last Day. And let a group of the believers witness their punishment.
Qur'an 24:2
Finally, it's likely that there will be serious social ramifications of this relationship; Islam encourages forbidding what is wrong (Qur'an 3:104).
You might also be interested in Advice to a Christian Man Who Wants to Marry a Muslim Woman, since Christian husband + Muslim wife marriages are also forbidden in Islam.
