The big difference in your comparison is that vaccines and the roundness of the Earth are very well established with overwhelming evidence supporting them. The heliocentric solar system was not well established at that time. That's what the whole discussion was about.
Copernicus' book had been published in 1543. Galileo was convicted of heresy in 1633. That would be like someone in 2043 being convicted of heresy for saying that DNA has a double helical structure.
Parallax, predicted by the heliocentric model, was only observed in 1806. Until then, its apparent absence was a strong argument against heliocentrism. (We just didn't know that stars are so far away.)
DNA structure is not a very good analogue because here we already had the X-ray image, and the question was to find a molecular structure that matched the data.
Also, the discovery of DNA fulfilled a prediction by Darwin's theory of evolution: that there had to be a mechanism by which traits were passed on to the next generation. It's something that made sense within the paradigm of the time, and it fit the observations.
Heliocentrism made sense on some level, but not on another, and it didn't fit the observations of the time. Only when Kepler made a model that fit observations, and Newton's theory of gravity explained why it had to be that way, did heliocentrism made scientific sense and was geocentrism obviously wrong.
It is not about evidence, it is about sure you are about your argument, and as it was stated below - Galileo wasn't the first one to propose heliocentricism.
But at the time, without robust mathematical concepts of inertia, forces, etc there were some reasonable arguments against heliocentrism. Plus, Galileo ignored Kepler's elliptical orbits, which are closer to reality.
