American Jewish leaders, who last week denounced Interior Secretary James G. Watt for his controversial letter to Israeli Ambassador Moshe Arens, are now seeking to quiet the calls for his resignation.

The leaders said they do not view the letter as grounds for dismissal and have privately urged Democratic politicians to put the matter to rest. Several expressed concern that the uproar could foster ill will toward American Jews.

"The last thing we need is another Andy Young," one said, referring to the tension between blacks and Jews following former U.N. Ambassador Andrew Young's resignation in 1979. Young left the Carter administration after holding an unauthorized meeting with a Palestine Liberation Organization official.

"I'd like to see the issue of the letter go away," said Hyman Bookbinder, head of the Washington office of the American Jewish Committee. "We've made our point, and we are extremely grateful that the White House immediately dissociated itself from the letter."

Leaders of the Union of American Hebrew Congregations, the American Jewish Congress and B'nai B'rith International expressed similar sentiments yesterday. All had attacked Watt for writing the letter, which warned that American support for Israel could be weakened if Jewish liberals opposed Watt's energy development program.

They criticized the message as a threat and called Watt "insensitive" for addressing his plea for Jewish support to a foreign ambassador.

Meanwhile, Sens. Daniel Patrick Moynihan (D-N.Y.), Edward M. Kennedy (D-Mass.) and nine other Democrats yesterday introduced a resolution calling on the Senate to denounce Watt's letter. The resolution did not mention resignation, but aides said this was an attempt to lure Republican cosponsors.

"People have to understand that this is cumulative. It may start with concern about the environment, but it's the combined effect of all of his behavior which put him in this position," Rep. Edward J. Markey (D-Mass.) said.

Watt has defended the letter, and President Reagan said through a spokesman he has no plans to discipline Watt.

Watt also came under fire yesterday for a letter which touched on Middle East policy that he wrote to Democratic congressmen opposed to his offshore drilling policies. The letter warned that American soldiers may have to fight "on the sands of the Middle East . . . if the policies of our critics were to be pursued."

In a speech on the House floor, Rep. John Joseph Moakley (D-Mass.) called on Reagan to take "immediate corrective actions by either restraining his Watt's comments with regard to U.S. foreign policy or remove him from office."