Abbas had previously said the Islamist group was responsible as it controls security in the Palestinian enclave, but Monday evening said they were "behind the attack". He further said that if the "assassination attempt" had succeeded, it would have opened the door for a bloody civil war.
Abbas's government has limited autonomy in parts of the West Bank, while the Jewish state annexed east Jerusalem in a move never recognised by the worldwide community.
Now there are at least 250 Israeli settlements in the West Bank.
Abbas added that Trump "views the settlements as legitimate", a reference to Israeli settlements on areas of land considered by Palestinians to be their own. Israel regards Jerusalem as its eternal and indivisible capital, although that is not recognised internationally.
Those steps, along with an Israeli-Egyptian blockade, have worsened Gaza's long-running humanitarian crisis.
The Palestinian Authority government on Tuesday called on Hamas to transfer full control of the Gaza Strip to it "all at once".
Hamas and Fatah agreed a reconciliation agreement in October but it has collapsed. Hamas seized government institutions and replaced Fatah and other government officials with Hamas members.
Hamas has controlled Gaza since ousting the PA in 2007 from the territory.
Hamdallah and Faraj were not hurt.
For his part, and in response to President Abbas words, Friedman suggested that Abbas' remarks amounted to anti-Semitism while speaking at a Jerusalem conference, later on Monday.
He said he would take new punitive measures against Hamas. Hamas called his comments irresponsible and said they aimed to "burn bridges and reinforce divisions". He said the Palestinian leader needed "to choose between hateful rhetoric and concrete and practical efforts to improve the quality of life of his people".
The serious rift between the two major Palestinian groups have emerged on the assassination attempt on Prime Minister shortly after US President Donald Trump's recognition of Jerusalem as Israeli capital.
The Special Rapporteur recommended that Israel "ensure regular and reliable access, at all times, for all Palestinian patients who require specialized health care outside of their jurisdictions, consistent with genuine Israeli security concerns" and to "comply fully with its obligations under worldwide human rights and humanitarian law with respect to fulfilling the health needs of the protected population", among other recommendations.