No. After he starts dating Ginny, Romilda Vane asks if he really has a tattoo of a Hippogriff and Ginny says no it's a Hungarian Horntail "much more macho", but that is simply her mocking Romilda. She told Romilda that Ron had a Pygmy Puff but didn't say where he had it. This set Hermione to burst with laughter.
He doesn't need a tattoo, he's got better. Hes got a lightning curse scar. No one can ever get a scar quite like his without mutilating themselves on purpose, or by surviving a killing curse and becoming a living horcrux.