Is is possible to have hearing damage because of stethoscope use?

I bought a new stethoscope and the sound is really louder than other kinds of stethoscopes that I used before and I'm happy about this. However, I feel a slight pressure on my ear (on my eardrum) whenever I press the diaphragm on the skin. I worried that I maybe unintentionally damaging my ear drum because of this. Is it possible that I can get ear damage because of stethoscope use?
The ear pieces on your new scope probably just seal really well. Which is why it sounds louder and why when you press on the diaphragm you feel slight pressure in your ears.

Did your stethescope come with an alternative set of ear pieces that might fit more comfortably?

The most that could happen is soreness in the entrance to your ear canal. Soft tipped ear pieces should help that. With the auto monitoring done these days, I haven't heard of this being a problem.