Look at the Crimean referendum, those who weren't ethnically Russian didn't vote for three reasons; voter intimidation, a rigged election and an unwillingness to give the vote even a fig leaf of legitimacy. If all the non-ethic Russians had voted it would have been used by Putin & co as proof of a fair vote. If an election is rigged then the only rational action is not to vote, but like I said before people are rarely rational despite most elections in the US being rigged.
So you are comparing an ethnic vote in a second world country to the United States. A country that hasn't been around in its current form for fifty years, to the longest running single democracy in the world. I think that's a stretch of the wildest sorts, but ok.
Voter Intimidation: The US has laws that have been on the books for decades making voter intimidation illegal. If you've ever noticed when/if you went to vote, groups are not allowed to even put up fliers near voting booths (I believe the current distance is 100 yards, but that's off the top of my head from memory of the last time I voted a month ago) Furthermore, the US has an institutional system set up to prevent it. We have police and the government itself. We also have the power of the press and right to assemble, which can be powerful motivational tools to spur debate and change.
Rigged Elections: The US has been, and is still, considered one of the most free countries in the word. (Freedom House's Ranking of the US is 'FREE', vs Ukraine's which is "PARTLY FREE' for starters). If you are charging that the actual elections, year after year, are rigged, then I would insist on proof. Proof using international criteria such as ballot box stuffing, multiple voted per person, etc. I won't say it doesn't happen at all, but the vast majority of elections are free and fair, and have been considered as such for many years. Otherwise it's an absurd accusation to generalize like that.
Unwillingness to give legitimacy: So, I'll ask you, who voted? Because I have tons of reports saying that Russian citizens voted in the election. Or that Crimean citizens were given Russian passports in order for them to claim Russian citizenship. How can you honestly call that legitimate in the first place when other citizens are taking place in a sovereign state's elections? When one country actively participates in another country's elections, how is that legitimate?
You want to compare that to the US? That'd be as if New Mexico sudden voted to secede from the Union tomorrow, claiming to be part of Mexico. And Mexican citizens came across to vote. Who would call that legitimate!?
The US has actual procedures set up to handle such things. For starters, we have a strong national court where grievances can be brought. And, in Texas v. White, SCOTUS struck down unilateral secession. We have a press that is able to move and print freely. We have, the possibility, of an informed electorate. We have freedom of travel. And, a strong central government made up of a legislator and executive branches. These things have grown over two hundred years into what they are now.
Ukraine has hardly any of those.
Democracy is an institution. It doesn't just happen overnight. Britain's government has been around for hundreds of years and they still have challenges (Though not in its current form). France is on it's fifth constitution and they have trouble. S. Korea went from a dictatorship to a free country and they still are wrestling with large issues. But they've also made great strides in each one of those cases. I would love to be pointed to a case study where a country has chosen the route of non-participation and such changes have been made. How did the populace come up with those changes? Who implemented them?
Personally, I think campaign finance and gerrymandering are two of the biggest challenges to elections at the moment. But, I'm not convinced that by not voting, it will change things. In fact, I see it as a tacit approval of the status quo. So, if there is a candidate who supports my positions on changing those, or who has ideas I support to change them, why would I NOT vote for them? Convince me of that if you want to win me over.
I've always wished we had 100% voter turnout. I certainly don't like only a third of the population choosing the candidates, and a majority of that third deciding the winner. But what seems to be the opinion of some is to whittle the turnout down to nothing, in the vein hope that "It'll show them!" Which seem fatalistic and apathetic to me. It's like someone who thinks they deserve a good job, without actually doing any work.