It's a fight we'll win, but you've got to accept it's a long fight. A very long one, measured in terms of generations. That's been going on for longer than the almost half century since the current movement started. Yes, we're making progress, but we're not even halfway there yet.
There're a bunch of European countries that made same-sex marriage legal, on par with het marriage, with the same rights and benefits, and even possible adoption. On top of my head, UK, Spain and the biggest one yet, France. Probably because at the time it came up, the governement of these countries pushed in that direction, despite the vocal opposition of a minority.
I think most "civilized" societies are ready for it, but the average het people won't cast their vote because of that issue, but rather because of economic concerns. For it to happen, there need to be a political will, or nothing will happen.
So, a Japanese lesbian couple could perfectly marry in France for example and adopt children there (though their marriage would not be recognized in Japan, yet).
States that allow same-sex marriage for further reference:
In Europe: Finland, Sweden, Norway, France, Netherlands, Spain, England, Wales, and Scotland but not Northern Ireland, Belgium, Iceland, Portugal, Denmark
The Americas: Canada, Brazil, Uruguay, Some states of the US and Mexico, Argentina
Everywhere else: South Africa, New Zealand.