Israel's announcement came one day after Palestinian militants killed six Israeli civilians during a bomb attack on a checkpoint in the Gaza Strip. The Israeli government contends the attack was launched from a Palestinian Authority base.
"Israel informed international leaders today that there will be no meetings with Abbas until he makes a real effort to stop the terror," Sharon spokesman Assaf Shariv said.
Palestinian Cabinet Minister and longtime negotiator Saeb Erekat said Israeli government officials contacted him on Friday to say Sharon was "suspending all contact."
Abbas was elected to the Palestinian Authority's highest office by a historic vote held last weekend. Abbas was a top official and adviser to longtime Palestinian leader Yasser Arafat, who died in November. Arafat's death cleared the way for new Palestinian leadership and fueled hopes for a new start in the Israeli-Palestinian peace process.
Sharon's administration and the U.S. government had severed direct links to Arafat and denounced him as a failed leader who fomented terrorism.
Abbas, scheduled to be inaugurated on Saturday, was reportedly angered that Sharon would hold him responsible for militant attacks before he assumed full authority.
The Palestinian president-elect has denounced militant violence and has said he will work toward a truce with militant groups. He has also vowed not to use force to reign them in, a stance that is an ongoing point of contention with Israel.
The Palestinian militant groups Hamas, Al Aqsa Martyrs' Brigade, and Popular Resistance Committee, claimed responsibility for Thursday's checkpoint attacks. The Al Aqsa group has ties to Abbas' Fatah political party.
Earlier Thursday, Israeli troops in Gaza reportedly shot and killed a Palestinian gunman and a man who was said to be driving his pregnant neighbor to a hospital. Israeli soldiers said they suspected the driver was a militant.
Israel responded with helicopter gun ship strikes on a Palestinian camp and by sealing off part of the Gaza Strip. News reports on Thursday cited Palestinian witnesses who said a medical facility was destroyed and one person was wounded.