Avigdor Lieberman was "barbaric" in his comments, Abbas adviser Nimer Hammad said, accusing Lieberman of being a "thug" and "speaking with a jungle mentality," The Jerusalem Post reported.
Hammad expressed "regret" Israelis voted for Lieberman and said, "The real obstacle to peace is not President Abbas or the Palestinian policy, but Israeli policies and Lieberman's arrogant statements."
Lieberman told reporters earlier Monday: "If there is one obstacle that should be removed immediately, it is [Abbas]. If he were to return the keys and resign, it would not be a threat, but a blessing."
Many other Western-trained Palestinians could replace him, Lieberman said.
Neither Israeli Prime Minister Binyamin Netanyahu's office not the White House had an immediate response.
Lieberman spoke in advance of a planned Wednesday meeting of the so-called Quartet of Middle East peacemakers -- the United States, the United Nations, the European Union and Russia -- amid media rumors Israel offered to halt settlement construction in the West Bank and East Jerusalem.
Netanyahu's office denied such an offer was made, the Post reported.
Palestinian officials have said they will not negotiate with Israel until it halts all settlement construction in territory Israel captured from Jordan in the 1967 war, where the Palestinians envisage their future state.