The Greens proposed an additional motion to formally censure the senator over his remarks.
But Labor sided with the government to reject the proposal, reasoning that it did not want the parliament to make Senator Anning a "victim".
"We have no intention of making Fraser Anning a victim. We have the absolute intention of both condemning this remarks and of taking on his arguments because they are wrong," Senator Wong said.
"I think today what the Senate has shown is that the best way to deal with division is to come together. The best way to deal with prejudice is to assert acceptance and tolerance. The best way to deal with people going low is to go high."
Greens leader Richard Di Natale said "hand-wringing and condemnation" was not enough.
"It’s incredibly disappointing that the two old parties have sided with Anning, blocking discussion of a move that would officially censure his inflammatory remarks," Senator Di Natale wrote in a statement.
