He won the electoral college vote. He lost the popular vote. This hasn't happened for 17 years. A vote in the middle states is more valuable than a vote in the coastal states which stretches the definition of democratic. The system isn't going to change anytime soon though because it benefits the GOP. Part of the reason for the protests is that the majority of American's didn't vote for him. Imagine if the EU referendum results were remain with the popular vote and leave with the constituency vote. You think people would have taken that well?
The forum is paid to quote across multiple pages so I'll also say that the women were not just protesting against democracy. They are protesting because they are angry, scared, revolted at a man who has said some disgusting thing about women which the rest of the country ignored and still voted for him.
The UK has a 'special relationship' with the USA and when article 50 is triggered the importance of this relationship will increase. Of course we are trading partners, which in turn means we are screwed: http://www.independent.co.uk/news/uk/politics/theresa-may-donald-trump-nhs-us-trade-deal-brexit-torture-a7548156.html
And again, just because I can't quote it, maybe people were protesting across the globe because the position of US president is one of the most powerful roles in the world and Trump will have a global impact. Maybe it's because there isn't female equality. Look at the wage gap, look at the backwards views on consent, look C suite board's gender ratios. Or maybe it's because they aren't completely heartless? When has anyone previously objected to transboundary protests on this forum?
Quotes again but Trump lies, much more so any Democratic candidate. He won because he pandered to the vote of the middle states, not for telling the truth. True he said things that no candidate would dream of but they were also what these states wanted to hear. FYI you can tell when he lies because he screams 'FAKENEWS' on Twitter.
Which brings me to why protesting was a great thing to do: Trump is incredibly thin skinned. Anytime someone proves him wrong or catches him in his lies he explodes. Look at the Russian incident. His press sec wants to ban press from the White House because they know when he is lying. He even outright denied crowd numbers at this inauguration. FYI that speech that Trump gave was in a memorial for fallen Americans. Only one other speech was given there and that was by Bush after 9/11. It was an incredibly offensive move by Trump.
Notice now how the photos published by the parks service of the inauguration were forcibly removed because they offended Trump.
He already has with the passing of the abortion gag: http://www.abc.net.au/news/2017-01-24/trump-makes-early-move-on-restricting-abortions-around-the-world/8206444
But it's correct. Straight white males are the most powerful group in America. Unless we see every other American riot this isn't going to chance. Minorities don't have guaranteed platforms and struggle to keep them over the next 4 or 8 years.
TLDR: The Right's biggest success was convincing people that they were the party of the people. The Left's biggest mistake was letting the Right get away with it.