WASHINGTON — President Obama’s new budget has opened a debate over what it means to be a progressive Democrat in an age of austerity and defines him as a president willing to take on the two pillars of his party — Medicare and Social Security — created by Democratic presidents.

By his gamble on Wednesday in proposing budgetary concessions to Republicans on Social Security, the 1935 creation of President Franklin D. Roosevelt, and Medicare, the legacy of President Lyndon B. Johnson, Mr. Obama has provoked angry supporters on his left to ask whether he is a progressive at all.

The A.F.L.-C.I.O. president, Richard Trumka, in a blistering statement, called the proposed changes “wrong and indefensible.” An e-mail from Representative Alan Grayson, a liberal from Florida, was headlined “President’s Budget Breaks Promise to Seniors.”

But to Mr. Obama, cost-saving changes in the nation’s fastest-growing domestic programs are more progressive than simply allowing the entitlement programs for older Americans to overwhelm the rest of the budget in future years.