It's okay to do a tub bath with a newborn. I did a lot of research on it as a student, and it is supported by studies. Tub baths are tolerated better by babies, help with temperature regulation and have no effect on cord healing when compared to sponge baths. Just don't let LO soak in the tub longer than it takes to wash them, and be sure to dry the cord thoroughly.
My LO was pretty happy in the water, and her stump came off at 9 days, no problems healing.