The simple answer is that a 35-year-old man doesn’t need to talk to his family about taking the next step in a relationship. But I think what you are really asking is how to help your family accept your girlfriend when they are worried that her depression may set you up for tragedy. If that’s the case, all I can suggest is that you tell your family members that you appreciate the love that is behind their concern and reassure them that you love her enough and are strong enough to manage the not-so-good with the good. Then simply ask them to respect your decision and to give your girlfriend an honest chance.
Hopefully, your girlfriend has gotten appropriate treatment and knows how to manage her depression. Hopefully, you can help your family get to know her for all the good reasons you are in love with her so they will understand why you want to make a life with her.
Do think carefully about who should move where. If your family is really against the relationship and won’t be welcoming, it may not be fair to ask your girlfriend to come to your country — at least not yet. Remember, if she moves, she won’t have her natural supports and will have disapproving in-laws. It might be advisable for you to go to the States for a time to solidify your relationship to the point that your family gets used to the fact that she is in your life. I don’t presume to know the right answer. I’m suggesting that you and your girlfriend take the time to really talk it through.
I wish you well.