Possibly, if you make two trips and don’t mind mad science and teaching. I think you need to make two trips regardless, but it’s possible you might not.
The first trip is to gather samples of all the communicable diseases you want to fix from the Europeans. This is important, because diseases mutate and change at a terrifying rate, so a viable vaccine for today’s influenza is not a viable vaccine for the influenza of yesteryear. You then need to bring them back to today for step two, which is:
Make a communicable vaccine. Mad science time, and the part of this most likely to go horribly wrong. As have been noted in a couple of other answers (Mass disease prevention: The feasibility of airborne vaccines and Vaccines spreading as viruses) there exist live-attenuated vaccines which can vaccinate and also spread. There are a couple of downsides but I’ll come to them. If you can create your own custom (heavily gm) version of this you can have a highly infectious but remarkably asymptomatic vaccine, in much the same way that cowpox, while infectious, vaccinated against smallpox. Given that nobody will have immunity to the virus it will spread like wildfire, not killing anyone but making sure everyone is safe when the actual killers show up.
Now, this can go wrong in a couple of ways. The first is the most obvious: your vaccine is a little too much and kills everyone anyway before their immune systems can kick in. Your vaccine needs to walk a super-fine line between being infectious enough to spread and being weak enough to not overtax the host, while also kicking off all the appropriate immune responses and creating the proper antibodies. This is not a small task. You may wish to bring some blood samples and/or test subjects from the past to the future to.. erm... make sure you get it right...
The second is less obvious and oddly, either much less deadly or absolutely deadly. Your vaccine will have every opportunity to mutate back into a lethal version of the disease. Now, in the best case this will kill a tiny fraction of people. Everyone else will have already caught and fought off the vaccine disease. In the worst case scenario the new disease mutates in such a way that the old vaccine is useless, and you just introduced the killer epidemic.
But this is where part three comes in! Education. Basically if your second trip to the past goes back a long enough way you can make sure the vaccine diseases have spread and also set up a known network of hospitals/health education centres. Get the mayans onboard with medicine. Teach the Incas about cleanliness and covering your mouth when you sneeze. Convince the priesthood that people who feel ill need quarantining for a little while, and get some idea of epidemic preparedness going on. A little knowledge can go a long way in situations like this. If even one chief realises he should stop ten ill travellers from travelling it might stop a potential epidemic (literally) dead.
Hopefully you’ll get lucky and be able to stop this, but all these steps are sort of longshots, some with the possibility of accidentally unleashing an even worse plague.
So please change the past responsibly.