I am a PhD student and I wrote a mail to an US prof for an internship offer, which will be completely funded by my parent institute (fellowship available) and hence the host US prof does not need to spend on anything. He replied in a matter of few hours that he has agreed to host me. I replied to him thanking him and also asking for an official letter from him that he has accepted my offer. He has not replied after that, it has been two days. The deadline is in a week. Do you think I should drop him a reminder or is it too soon?
Two days is no time at all. But if your first request for a letter didn't emphasize the time constraint then you could send a note apologizing for that omission and giving the deadline.
And if the letter was supposed to be addressed to the institution and not to you personally, it is possible that it has happened or will happen without your knowledge.
But if you were initially clear about the time requirements then a note after two days is just noise. The prof may be busy, even busy composing the note you need.