<I have accused you of nothing.>
Fair 'nuff. I was anticipating.
<I have acquaintances who I suspect are homosexual>
Acquaintances, not friends. If they were your friends, if you were theirs, you would know this about them.
<I give them the benefit of the doubt.>
What doubt? What benefit accrues to them as a result of the doubt?
<I don't know how old you are nor do I know whether you have changed your mind on the issue over the years>
I gave you a hint about my age when I said I was working fifty years ago. Do the math -- you'll be close. And yes, of course, I've changed my mind about this and many things over the years.
<...how a core teaching of the Catholic Church can be rejected>
The core teachings of the Catholic Church are found in the Creed; it's preposterous to describe the teaching about birth control, homosexuality etc. as "core teachings." 
"...can claim to be a Catholic."
I don't claim to be a Catholic; I am a Catholic. So are most of my gay friends, even the married ones.