According to the U.S. Army, he did save 75 people, although I have not seen the movie. Here is the official Army page on it.

Over the span of several hours, Doss treated the injured and, one by one, dragged them to the edge of the cliff and lowered them to safety in a rope sling. After each successful delivery, he reportedly said, "Dear God, let me get just one more man."

By nightfall, he had rescued 75 soldiers, including many of the men who had berated him earlier in his military career.

He also recieved the Congressional Medal of Honor, and the Tennessean newspaper published the citation for that which is kept by a historical society. That article is here.

Now, that citation does not mention 75 soldiers being rescued explicitly, but it does say

As our troops gained the summit, a heavy concentration of artillery, mortar and machinegun fire crashed into them, inflicting approximately 75 casualties and driving the others back. Private First Class Doss refused to seek cover and remained in the fire-swept area with the many stricken, carrying them one by one to the edge of the escarpment and there lowering them on a rope-supported litter down the face of a cliff to friendly hands.

... it continues with a lot of other acts of bravey after this, however