What should a Muslim do who fears harm when traveling to countries that permit homosexual marriage, if asked about his stance on this marriage, and he does not support it, and sees danger to himself in declaring non-support, and in declaring support, the effacement of his religion and his identity?

It is advised not to travel to non-Muslim countries for study, especially for women, due to the risks this poses to one's religion and self. One should emigrate to a Muslim country if they fear for their religion. However, if there is a compelling need to travel, such as for a specialization unavailable elsewhere, and you hope for safety for yourself, then there is no harm, provided you do not lie or approve of forbidden acts, and you travel with a mahram.