Yes, baseball is still America's past time, because it is still loved and popular among Americans. All children play t-ball. Everyone calls baseball players the boys of summer. Each year, people hope their team is in the world series. Baseball is a sport that is uniquely American, and as such has earned the name, America's past time.
Baseball is hardly a sport in my opinion. Half the game is spent sitting down except for the few chances at bat. The other half is spent sanding around waiting for your chance to touch the ball. That said the most true America's past time would be american football. It is so unique in that we are the only ones in the world who play it and we make its championship almost a holiday.
If any sport can claim to be solely America's, it's American football. Football is one of the most bizarre sports in existence, and one in which only the USA plays. Japan is by far more crazy about baseball than America is, so if anything, they could claim that baseball is their past time much more plausibly than America can.