In this scripture refrence you have to start at the beginning, see verse [18]and continue on through to verse [32]to see that there are many sins leading up to these folks being turned over'or being given up'or as we would say I just give up on you.These are people who knowing the judgment verse 32 not only do they do these things but they have pleasue in those who do them.So I believe that you can not be a christian and partake of the things that God calls abominable just as one who calls himself a christian can not murder or commit adultry.They must repent and turn from their wicked ways to follow christ.This scripture describes a terrible place to be.When God has given us up or allowed us to go in the ways we've chosen. 

God gave them up-abandoned them for their lusts and idolatry of images made...[v.21-]They were fools and God abandoned them....and NO...you cannot be a homosexual and a christian at the same time. Homosexuality is an abomination and He says that it is immoral and evil. Now, if that person wants to be a christian that person has to completely change and repent and ask Jesus into his/her life. 

[Romans 1:18-32] is actually an account of wrath and judgment. Homosexuality is a curse and a punishment. It is describes as 'unnatural' and a 'perversion'. The same people are listed along with 'evil, greed and depravity. They are full of envy, murder, strife, deceit and malice...'
I think that's enough hard matter to swallow for now. You can do a more thorough study of the chapter on your own if you like.
For a finishing touch, here's the last verse of chapter 1:

32 Although they know God's righteous decreee that those who do such things deserve death, they not only continue to do these very things but also approve of those who practice them.

Those things are deserving of death.

It is, however, possible that such can become christians. Everyone can become christians, and I'm sure we all know why. God's grace, mercy and salvation extends to all who would receive it and live in it. Once becoming christian, however, repentance and being born again comes into the picture and such things would have to be permanently discontinued.
What happens with if they are? Well, the curse resumes, and the chap who just got pulled up the cliff slips down it again. If he doesn't scramble up soon, he'd fall all the way to the bottom. Salvation is not under warranty if you deliberately mock it. That's the same as asking for forgiveness from your father and then slapping him in the face and jeering at him.