Yes you can. In fact a major airline has a 1 eyed pilot. he obtained his FAA medical by applying through some obscure FAA rule that says that even though you have medical problem, if you can successfully prove you can do the tasks required you can get the liscence. i think it called something like medical competency.
In this guys case he had to get a FAA doctor who is also a pilot to go fly with him so he could prove he could preflight, takeoff, fly, land ect.
He had an ATP MEL so he rented a twin so he could prove single engine capability.