In presence of a mahram a physician my see what is necessary of the 'awrah of a woman
First you must be aware that in presence of a mahram a phyisican is allowed to check a woman. As explained with more details in my answer on Had to show my full arm to a man. This actually sets your argument of a man seeing your wife's privat parts void. I remember the moments of birth of each of my sons (I still have a picture in mind for each of them) and I'd recommend being present for this and helping the wife and giving her comfort as else she would be surrounded by strangers. The only moment when I was sent out was when she got the epidural and that was only in one case the second time I was just asked to look away if I would fear seeing it.
The presence of a gynaecologist is generally not necessary during the birth
Secondly the presence of a gynaecologist is normally totally unnecessary for that act of birth unless there are any complications. In Germany for the usual birth there are two or three people present the woman who gives birth, the midwife who assists, helps and guide her and the father of the child to give his wife comfort etc. A physician is only called at the end or even after the birth to check the newborn and the woman who gave birth and probably sew the perianal rupture.
Sadly since pregnancy and birth became a business and many women prefer to have caesarean instead of a natural birth or one with the aid of peridural anaesthesia the presence of normally needless physicians became more and more a luxurious must.
It is recommended to have children and frowned upon or even haram to avoid having them
Thirdly it is rather recommended to have children and frowned upon or even haram ( in case you are afraid not being able to be in charge of them or afraid from poverty) -if it is possible- to refuse having them. While it might be necessary if there are any medical indications. See for example islamqa #127170 where it says:
It says in a statement of the Islamic Fiqh Council belonging to the Muslim World League: The Islamic Fiqh Council affirms unanimously that it is not permissible to limit the number of children in general and it is not permissible to prevent pregnancy if the reason for doing that is fear of poverty, because Allah is the Provider and Owner of great power, and there is no living creature on earth but its provision is due from Allah, or if that is for other reasons that are not acceptable according to sharee’ah.
As for using means of preventing or delaying pregnancy in individual cases where real and certain harm will result from it, such as if the pregnant woman will have to give birth in a manner other than that which is usual, and she will have to have surgery to bring the child forth, in which case there is nothing in sharee’ah to prevent her doing that. The same applies if it is delayed for other legitimate shar’i or health reasons confirmed by a trustworthy Muslim doctor. Contraception becomes a must in a case where it is proven that harm would result for the mother or there is the fear that she may die, according to the opinion of a trustworthy Muslim doctor. End quote from Fatawa Islamiyyah, 3/200.
and on a similar topic fatwa islamqa #7205.
Not all hospitals have only male stuff
Last but not least as said in my comments you may find hospitals where the majority stuff of the gynaecological hospital is female. This also should set your argument void to some extent. And you should inform yourself about the hospital and the process of birth before going there. In Germany all gynaecological hospitals have an information evening once a month where to be parents can seek information and learn about the hospital and stuff. Also note that not all physicians in non-Muslim countries are solely non-Muslims.
Epilogue and conclusion
Finally the view or attitude of Islam for life and giving birth is positive this means Islam even if it considers itself the last and final -true- revelation it encourages life and enjoying life as is and it doesn't encourage giving up life or having any negative attitude against life. Therefore marrying is encouraged, having children is encouraged and even if it might be the last hour labouring is encouraged:
... What has happened to these people that they say so and so, whereas I observe prayer and sleep too; I observe fast and suspend observing them; I marry women also? And he who turns away from my Sunnah, he has no relation with Me.
(The statment made by our prophet () taken from the long hadith of those sahaba who wanted to be as pious as the prophet (), see for example in Sahih al-Bukhari and Sahih Muslim)
"Marry women who are loving and very prolific, for I shall outnumber the peoples by you."
(See for example in Sunan abi Dawod, Sunan an-Nasa-i and Sunan ibn Majah)
"If the Final Hour comes while you have a palm-cutting in your hands and it is possible to plant it before the Hour comes, you should plant it."
(See in al-Bukjhari's Al-Adab al-Mufrad and also in Musnad Ahmad).
So from all of this you should know that your fear is not a reason that has enough weight to allow you to reject having children. Because you may choose a hospital where the chance of meeting a male gynaecologist is low or lower, you may inform yourself before the day of birth about the hospital and stuff, you may even consider travelling to a Muslim country or a country/location where your wife would rather be surrounded by female stuff ... so there are many walk around to solve the issue which was in first place created in your mind.
And finally be aware that staying -without necessity- or to settle in non-Muslim land is not appropriate for a Muslim based on:
I am not responsible for any Muslim who stays among polytheists. They asked: Why, Messenger of Allah? He said: Their fires should not be visible to one another.
(See in Sunan abi Dawod and Jami' at-Tirmidhi)
While it is rather allowed to go there for a purpose and come back to Muslim land.
