Homophobia certainly isn't 'responsible' for religion, that... doesn't even make sense, but homophobia transcends all of the major world religions today, and there have been examples of homophobia in societies that either preceded the modern, common religions, or were even relatively untouched by those religions. Even in the modern day, you have atheistic communities that are deeply homophobic (
link and
link).
The Philosopher/Psychologist William James theorized that revulsion to same-sex intimacy between men was something instinctive for humans, and that societies that had become comfortable with it became comfortable out of habit. James was writing in the 19th and early 20th century, so obviously society has changed, but arguably, this theory would suggest that religious arguments against homosexuality could have been informed by an instinctual predisposition against homosexuality.
With Homophobia, religions can be a block to progress (I'd argue that progress is a less homophobic society), but they don't have to be. Similarly for several hundred years, Western slave owners used a perverted Christianity to justify slave ownership, and yet, abolitionists also used Christianity as the foundation of abolition... And, of course, many Civil Rights advocates were deeply religious and bolstered their righteousness with Christian teaching. You see similar arguments made today by universalist and progressive churches (as well as progressive members of more moderate churches like Catholicism) where Christian teachings are used as an argument to love and build community. What this shows me, at least, is that religion is typically used to foment a societal/communal belief, and not necessarily the other way around.
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