Isildur became invisible too. From Unfinished Tales, the Disaster of the Gladden Fields chapter:
Isildur turned west, and drawing up the Ring that hung in a wallet
from a fine chain about his neck, he set it upon his finger with a cry
of pain, and was never seen again by any eye upon Middle-earth. But
the Elendilmir of the West could not be quenched, and suddenly it
blazed forth red and wrathful as a burning star. Men and Orcs gave
way in fear; and Isildur, drawing a hood over his head, vanished into
From this we can tell Isildur became invisible ("was never seen again by any eye upon Middle-earth") but in contrast the Elendilmir was still visible, until Isildur's hood was pulled over it. This then explains why Isildur was spotted and shot by Orc archers once the Ring had slipped off his finger. Interestingly, the only real difference between UT and the movie is that while Isildur was spotted swimming in the movie, he was shot while leaving the river in UT.
For those, like me, who didn't know what the Elendilmir is, it was a crown of the Kings of Arnor, containing a white crystal of Elven origin.
Gandalf also implies that all Men (and Hobbits) would become invisible, saying in LotR:
A mortal, Frodo, who keeps one of the Great Rings, does not die, but
he does not grow or obtain more life, he merely continues, until at
last every minute is a weariness. And if he often uses the Ring to
make himself invisible, he fades: he becomes in the end invisible
permanently, and walks in the twilight under the eye of the dark power
that rules the Rings.