I'm a bit of mixed up on this subject. Does asymmetric encryption like in RSA, ensure message integrity without hashing the message with an HMAC, nothing else is required, or is hashing necessary only when utilizing symmetric encryption?
No, RSA by itself definitely does not ensure integrity. (i.e. it's malleable.)
If the ciphertext c = me (mod n) is modified to c1 = c * t (multiplication), where t = xe, then the decryption of c1 will be
c1d = (c t)d = (me xe)d = ((mx)e)d = mx (mod n)
so the original message got multiplied with a number controlled by the attacker.