Even the Athenians, who founded democracy after all, understood that holding multiple votes on an issue was far from undemocratic. It's a complete and utter lie, and a misunderstanding of democracy, to claim that multiple votes on the same issue is somehow an affront to that principle.
Jackdaw mentioned Switzerland's plebiscites as an example. They, too, sometimes vote on the same issue multiple times -- albeit usually separated by more than a couple of years. Still, one specific example ought to be an illustration of the point. The Swizz voted on the issue of whether or not women should be allowed to vote. The first time they tried this, in 1959, the proposal was resoundingly rejected. I should think most people would have been horrified if this were regarded as the end of the issue; luckily for us all -- and not least the Swiss women -- that result was overturned 12 years later.