That's a common misconception/lie/tactic. It's not their private lives that we are talking about, but rather the public policy/law that they want to change/impose.
Yes, but it's generally public policies that only affect them.
Honestly, is a gay couple getting the legal recognition to be able to visit their partner in the hospital, handle medical consent forms, inherit their estate, etc, going to have any affect on you (outside of mental frustration which is a personal problem which can happen to anyone easily irritable) at all?
We're not talking about socialized medicine that's going to force us onto a public health plan here. We're not talking about a welfare state that is going to increase our taxes. We're literally talking about just letting them have the option of setting up the same contract that the rest of us do.
Outside of a religiously based hangup (which constitutionally cannot have any recognition in our laws), there is not reason why it would affect anyone else.
Now as to military chaplains being forced to perform these, I'm of mixed opinion. IMHO, if they're paid employees, then they should have to perform any legal ceremony that the state recognizes. If you're a paid employee that that's part of your JOB. That said, to me, its a conflict of interests to have an official chaplain in the military. Being of religious nature, that type of thing should be left up to non-official community organized groups (either soldiers setting up their own services in their spare time or missionaries). That can be donation supported or just volunteer missionary work. Being autonomous groups with no official standing (and with the religiously rooted marriage ceremony being legally removed from the equivalent legal arrangement in civil law as I suggested), those groups would be free to marry or not marry whoever they wanted.
Put bluntly, whether or not "god" agrees with a decision or not should have absolutely no legal bearing on a situation whatsoever, because not everyone (and it's a rapidly growing not everyone - lack of religion is the fastest growing "religious" affiliation in the country) believes in that, and they all pay taxes too.