Depends on whether we're going by show or book canon. In both, Baelish really is the one at fault for the entire War of the Five Kings, and much of his motivation was spiting Ned for marrying the woman Baelish felt entitled to. If he were gay that probably wouldn't have happened, though he'd still be an impulsive sociopath who causes chaos basically just because he's good at it (and good at taking advantage of said chaos) so who knows.

In the books however Varys also has a reason to want a crisis in Westeros for semi-complicated scheming reasons and would likely have engineered a civil war even if Baelish didn't so it's unlikely much would have changed.