prohibition of polygamy most certainly doesn't correlate with democracy.
Polygamy was prohibited in the entire Christendom throughout Dark Ages and absolute monarchies - not exactly the best circumstances for Democracy. Same with USSR and Nazi Germany.
On the other side of the matter, modern western democracies and USA are fairly obviously evolving into societies where polygamy will soon be legal (never mind practiced - what with French and their fabled stables of mistresses :))). I don't mean to sound like a homophobic zealot, but looking at things from legal standpoint, if you legalize marriage between ANY non-monogamic heterosexual couple (like, say, legalizing gay marriage), there is very little legal ground not to legalize any other marital arrangements.
As a matter of fact, BDSMish subcultures practice poly* arrangements nearly openly these days.
As to whether prohibition of polygamy is necessary for democracy, that is less clear. I'm not aware of any research supporting or refuting this, but a fairly plausible argument can be made that any culture which has polygamy (and not polyamory) might be LESS likely to be democratic in historic context, since in a true democracy, with average male/female distribution, the non-mated males would at some point organize and vote in the prohibition (unless they are explicitly excluded from franchise).
Also, as a side note, please don't confuse democracy with things that kinda look like democracy (hint: Egypt may have had elections, doesn't necessarily make it a democracy, 95% of the power is split between the military and Muslim Brotherood. Oh, and USSR used to have those nice democratic elections all the time).
Also, don't confuse "democracy for some" with what you may think of as democracy. While franchise restriction always exists and will exist (e.g., any society would likely stop convicted killers from voting), you can for example have extremely-limited-franchise democracy like city states of Ancient Greece (VERY VERY few people were entitled to a vote, none of them women, and few of them with no access to mates).