Being an introvert myself (yeah I know what youre thinking), I seem to have no problem talking to patients for a few minutes to elicit a history. My interest in clinical medicine, and wanting to come to a diagnosis, is enough to get me to talk for a while.
I feel the same way. I consider myself extremely introverted, like the OP probably does. So much so that I often wondered, early in my training, how I could possibly meet and interact with dozens of strangers every day. However, I'm in a very clinical field, and it's nowhere near as bad as I'd feared. It's not like you're expected to make a lot of small talk, which I am still not very good at. Getting a detailed history and doing a focussed exam are somehow not as taxing. Talking to patients about test results, differentials and prognoses just seems natural after a while. I would guess the patients I see consider me somewhat detached and businesslike, but I answer all their questions and address their concerns, and that's what's most important to them, I'm sure.
That said, I still have a bit of a hard time walking into a room where there are 10-15 family members around the patient. If I can, I try to come back later when there are fewer. I suppose you could always politely ask the non-essential visitors to step out for a few minutes, citing patient privacy and comfort (which is probably a legitimate concern).
And contrary to what some previous posters feel, I actually find return visits in my clinic easier to deal with socially than new patients or new consults in the hospital. They're already familiar with your personality and bedside manner, and the appointments are shorter and more focussed. It appears that most patients are fine with me, so being introverted doesn't necessarily translate into being a bad clinician.
But yes, I do cherish the non-interaction time, spent writing/dictating notes, reading studies and charting, and that's at least half of your day.