A State Department official, who spoke on condition of anonymity to discuss visa issues that are normally confidential, disclosed that Abbas and other officials from the Palestinian Authority (PA) were among those affected by new visa restrictions.
Palestinian representatives assigned to the UN mission, however, were granted exceptions.
The move is the latest in a series of steps the Trump administration has taken to target Palestinians with visa restrictions and comes as the Israeli military declared Gaza's largest city a combat zone.
The State Department also suspended a program that had allowed injured Palestinian children from Gaza to come to the US for medical treatment after a social media outcry by some conservatives.
The State Department said in a statement that Rubio also ordered some new visa applications from Palestinian officials, including those tied to the Palestine Liberation Organisation (PLO), be denied.
"It is in our national security interests to hold the PLO and PA accountable for not complying with their commitments, and for undermining the prospects for peace," the statement said.
It said that to be considered partners for peace, the groups "must consistently repudiate terrorism, and end incitement to terrorism in education, as required by US law and as promised by the PLO".
The Palestinian Authority denounced the visa withdrawals as a violation of US commitments as the host country of the United Nations and urged the State Department to reverse its decision.
It said in a statement the Palestinian presidency "expressed its deep regret and astonishment" at the visa decision, which "contravenes international law and the Headquarters Agreement, especially since the State of Palestine is an observer member of the United Nations".
UN spokesman Stephane Dujarric said the world body would be seeking clarification from the State Department.
"We obviously hope that this will be resolved," he said. "It is important that all member states, permanent observers be able to be represented."
The State Department said representatives assigned to the Palestinian Authority mission at the United Nations would be granted waivers under the US host country agreement with the UN so they can continue their New York-based operations.
The Trump administration has pursued a crackdown on some of those who have legal permission to come to the US, and at times the standard for releasing once-privileged information on cancelled visas seems to have been relaxed to make a public point.
For instance, the State Department's No.2 diplomat posted on social media when the US pulled visas for British punk-rap duo Bob Vylan after they led crowds in chanting "death" to the Israeli military.
The Palestinian ambassador to the UN, Riyad Mansour, told reporters Abbas had planned to lead the delegation to the UN meetings and had been expected to address the General Assembly, as he has done for many years.
He also had been expected to attend a high-level meeting co-chaired by France and Saudi Arabia on September 22 about a two-state solution, which calls for Israel living side-by-side with an independent Palestine.
