Can online voting save democracy?

In theory yes. But once again how do we know whether or not such votes were counted properly?

No. It is not voting that matters, it is thinking about voting. Successful democracy requires that people vote with care. Online voting, while helpful for those who have difficulty with other forms, will also make it easy to vote in a casual, unthinking way.

Then we would have people who would cheat (gasp!) the system and cast 1000 or even 1000000 votes for themselves and of course the brains behind the fraud would simply vanish into the night never to be heard from again. Meanwhile we get a scenario from the BBC series, spooks or MI5 depending on which side of the pond you live. Brutal suppression of the masses, a one party system, wholesale torture and so on..
Save Democracy? No...more like instant destruction of Democracy..
I would far more prefer leaving the polls open a full 24 hours to allow shift workers the opportunity to vote on election days. I missed several elections because I had to work to many hours..Had to get to work at 6am and didn't get off till 8pm after the polls closed..

No.

The goal in our republic should not be to simply get as many people as possible to vote. If that were to happen, you would have people who don't really care and don't research the issues voting. That is a bad thing.

What you want is to increase the amount of people who CARE enough to think about the issues and do some research, especially about their candidates and issues on the local level. Then, if they care enough, they will get off of their duff, go down to the local precinct, and cast their vote. This is not too much to ask in a representative republic, like we have. The less electronic gadgets involved, the better, as far as I'm concerned.

What legitimacy would people have that their vote counted? I guess those people in Washington would love this idea!

Not in a Republic.