Correlation does not imply causation.
Most modern states have monogamy.
Even more "most" modern democratic states have monogamy as a result of direct influence of Christianity (Christian Europe or its former colonies).
So, there's a definite correlation, but due to having a clear historical explanation, it offers no need to seek causality.
Having said that, the causality may very well be there as well, but it would be extremely hard to prove due to dearth of data points and ability to construct independent variables. One can of course spin theories.
A popular belief (seemingly expressed even in the question) is the theory that, in non-monogamy with equal demographic sex distribution, the non-paired males, if given vote, would of course vote to prohibit nonmonogamy, as it would raise their chances of obtaining female mate.
However, this gets trickier if you go for modern democracy where franchise is (unlike the case with Athens or pre-1900 USA) extended to women as well - since women generally are considered to have a preference for hypergamy which may correlate strongly with nonmonogamy. This is politically sensitive enough that not a lot of decent research is done on the topic.