Short Answer: Absolutely not. There are no conditions under which a person is doomed to Hell despite having a sincere desire and willingness to return to God.
This will be tricky to explain, so for the sake of space, I will link passages rather than quoting them. I will address the common counterarguments to my answer at the end.
Nature of God
This topic cannot be understood outside of an understanding of God's nature.
In other words, He is perfectly willing and able to help anyone return to Him - this was just as true for a sinner like Hitler as it was for a sinner like Saul. But there is another important variable.
We know that God desires all men to return to Him and be saved, and Satan desires all men to reject God and perish, but we know neither of these desires will come to pass. This proves there is another variable - us.
God is omniscient; He knows everything - even our hearts - even before we are born. That is to say, God knew whether we would ultimately embrace Him or reject Him before we even existed! It is according to this foreknowledge that He "chose" us, deciding who He would draw to repentance.
"Doomed Forever to Hell"
The key thing to understand about Heaven & Hell is that ultimately, most people don't want to return to God. We will all "live" forever, but if we ultimately decide to reject God, our eternal "life" will be eternal separation from God (also known as "Hell"). I think it's less about God damning and dooming people, and more about people rejecting Him and God giving them what they wanted, leaving them alone.
Everyone is born sinful - this isn't a problem unique to homosexuals. Homosexuality is a sin, and believers are called to reject and overcome such carnal impulses. As far as the idea that they can't overcome it because of genetics, all I can say is be careful who you learn from - the world may say that homosexuality is genetically predetermined, but they also say water can't be miraculously turned to wine, there is no God, Jesus is a liar, etc.
Unable to Understand
Everyone is born in a state where they are unable to understand God's ways. In terms of physical inhibitions, I think you would enjoy reading up a bit on Helen Keller - before the age of 2 she became deaf and blind, but was still able to come to God in a beautiful way (as I understand it.) Regardless, there are two important points to make here:
God doesn't ultimately even need a person to be born to know their heart, what (if anything) would bring them to repentance, and in which eternal dwelling to prepare a place for them!
Despite Point #1, God will never condemn a person who is legitimately ignorant and hasn't had an opportunity to respond. Remember - He is perfectly just.
Ultimately, we have to recognize that it would be ridiculous to think our God would send someone to Hell for eternity who was unable to choose Him. It is simply inconsistent with His nature as revealed in scripture.
Answering the Objections
There are some popular "counterarguments" to this teaching, which can be summarized as:
Willful rejection of God: For example, "blaspheming the Holy Spirit" results in not being forgiven, and denying Jesus before men results in being rejected by Jesus before God.
Willful rejection of God's ways: For example, refusing to forgive others results in not being forgiven by God, and continuing to willfully practice sin results in there no longer being a sacrifice available to you.
Let me emphasize once again, that I am using the term "willful rejection" to refer to an "informed desire and decision to ultimately reject". This is the only point important to salvation in scripture - at no point are mistakes, ignorance, etc. met with unmerciful condemnation despite repentance!
So all of these counterarguments are really the same, and could be summarized as "if you ultimately choose to reject God you can't be saved." Of course that is true! But it is irrelevant to your question. You are asking if it is possible for a person to "willingly" desire to follow God, but be unable to do so because of a past mistake. The answer is definitely no.
To say it a different way, of course a willful rejection of God and/or His ways precludes you from salvation! But then again, such a person would not be seeking repentance... if they did, it would be evidence that the so-called "unforgivable sin" they committed was not "willful" in the manner I've been describing, and was thus not "unforgivable"!
God will give every opportunity necessary to a person in order to draw them to Himself if they would ultimately be willing to chose Him. To argue otherwise would be to undermine God's just predestination according to foreknowledge. (I hope that is clear.)
Since the blasphemy argument got some attention here already, I'll quickly speak on it. This article covers the topic pretty well, but briefly, this is not a slip of the tongue or mistake which precludes you from true repentance, but rather, it is evidence of a heart in willful rejection of the very presence of God. Of course, if a person "blasphemes" out of ignorance, that can be overlooked upon repentance;
"even though I was formerly a blasphemer and a persecutor and a violent aggressor. Yet I was shown mercy because I acted ignorantly in unbelief" - Paul