I think it's true that Martians will feel a great cultural separation from Earthlings (note to self: think of a better demonym), but I don't think it's true that colonies have historically become disinterested in the politics of their founding countries (be sure to consider only colonies that establish new populations shipped in from a host country rather than subjugating natives, since AFAIK Mars has no natives to subjugate).
> No one born on Mars is going to have any reason to care
> about what happens on Earth any more than the people of
> one continent care about the day-to-day life of people on
> another continent.
But humans on Earth care about what happens on other continents incessantly. Today in the news: attacks on ISIS, Hong Kong protests, and the ebola outbreak. I care about all of these things, and as a taxpayer my money is influencing all of these things, though I cannot say that I was born in Iraq, China, or Senegal.
You follow big news events happening elsewhere, but that is far different from caring about the day-to-day life of people. Mars would have a CNN equivalent to broadcast what is happening on Earth as well. But how much of the international news focuses on the impact back home? We're worried about ISIS because we might have to send troops to fight or risk them somehow endangering us directly (attacks in other countries, long range missiles, etc). We're worried about the ebola outbreak because there is potential to spread to our own nation. We're interested in the Hong Kong protests because of the impact that China has on the international economy. Watch CNN or Fox or Sky or whatever and listen to how often they will say "And what is the impact to [insert your country]?" or "How long until we're facing this threat in [country]?"
What on Earth (literally) could threaten the Martians once they've become self-sustaining? The only time they would care about what happens on Earth from anything more than a pure curiosity standpoint is when they are dependent on Earth for survival. After that point, until we develop ships fast enough that we can get to Mars as fast as we can get across the Atlantic and trade once again becomes an issue, who cares what happens to Earth? Mars is just fine.
