I would find it suspect if my specialist was googling unless I had something really weird going on, and might consider going to a different specialist.
If they're a neurologist and Googling "what does brain do", sure.
If they're searching up safety data on a slightly-over-recommended dose for a particular medication (like, say, a gastric bypass patient who takes an oral med that's not absorbed as much as in a person with a normal stomach), I'd much prefer they be certain than guess.
I still routinely Google some things that "I should know" when doing software development. Despite being already correct 99% of the time, an internet search helps with two things:
* Presenting potentially new information or methods of doing something. My "old" knowledge might still be correct, but a "new" method may have optimizations or benefits.
* Even though, I'm already 99% correct, it's not worth the hassle of being wrong the 1% of the time - especially when the solution is 20 seconds of internet search.
If they're searching up safety data on a slightly-over-recommended dose for a particular medication (like, say, a gastric bypass patient who takes an oral med that's not absorbed as much as in a person with a normal stomach), I'd much prefer they be certain than guess.