If pens required gravity to pull the ink to the tip, then you wouldn't be able to write upside down or sideways. Perhaps you could perform an experiment?
If you were to do so, you would find that pens do indeed work (ish) upside down and sideways, and therefore do not require gravity for their operation. However, they don't work particularly well (you have to shake them quite often), and there is more to writing in space than just the microgravity environment.
Gel and felt-tip pens are optimized for use based on the viscosity of the gel/ink, which depends on the temperature and pressure of the environment in which they are being used. Therefore, they aren't particularly dependable in the wide variety of conditions which exist in space, so a better alternative was designed.
So to answer your question, normal pens could be used, but specially designed "space pens" are better.