OK--let's assume your premises are correct.
IF it could be demonstrated that same-sex marriages would result in more "stable relationships" -- would you still be opposed?
Straight couples have a very high divorce rate so apparently marriage has very little to do with "outlawing unstable relationships".
Suppose 99% of all same-sex partners have no interest in having or adopting children. Would you oppose same-sex marriage simply because 1% might express interest in raising children---even if they met all the same standards and requirements as a straight couple?
It is up to you to prove your case to the majority, not the other way around. So far you haven't done that.
Why just couples? Would you be willing to support polygamy being legal?
Or would you simply support the end of government interference into relationships altogether?
The divorce rate may a result of divorce being too easy to obtain. You can't blame the institution for the recent rise in divorces when they became easier to get. The institution has been around longer than easy divorces.
The normalization of gay culture will send a message to my kids that it's OK to experience sex with other (usually much older) men. This is why gays are trying (in England anyway) to lower the age of consent to 14. Imagine a 40 year old man or pedophile having legal sex with a 14 year old. Thats what they want for my kids. Their marriage aspirations are a way for gays to beleive that it's OK, normal to be perverted.