"I almost fell off my chair when I heard the president's remarks last night," he said on CNBC's "Fast Money."
"He essentially fired Ben Bernanke on the spot and gave him a fairly tepid testimonial afterward. It's time to really now focus on who the next chairman might be."
In an interview with Charlie Rose that aired Monday, Obama said, "Well, I think Ben Bernanke's done an outstanding job. Ben Bernanke's a little bit like Bob Mueller, the head of the FBI, where he's already stayed a lot longer than he wanted or he was supposed to."
(Read More: 'Full-Speed Ahead' Before Fed Meeting)
Meyer, who is senior managing director and co-founder of Macroeconomic Advisers, said it has "never been more important to have continuity."
"The next chairman will have a very challenging situation," he added. "It's unclear what the president wants."
Meyer, who served as a governor of the Federal Reserve from 1996 to 2002, also acknowledged a bit of an overstatement on his part.
(Read More: What the Fed Is Likely to Do)
"There's always a bit of hyperbole when I talk, but the president said he stayed longer than he wanted to and was supposed to," he said. "What does that mean? I'm just befuddled."
Trader disclosure: On June 18, 2013, the following stocks and commodities mentioned or intended to be mentioned on CNBC's "Fast Money" were owned by the "Fast Money" traders: Guy Adami is long C; Guy Adami is long GS; Guy Adami is long INTC; Guy Adami is long MSFT; Guy Adami is long AGU; Guy Adami is long NUE; Guy Adami is long BTU; Guy Adami's wife, Linda Snow, works at Merck; Brian Kelly is long US dollar; Brian Kelly is long Canadian dollar; Anthony Scaramucci is long AAPL; Anthony Scaramucci is long GS; Anthony Scaramucci is long WFC; Anthony Scaramucci is long MSFT; Anthony Scaramucci is long GOOG; Anthony Scaramucci is long INTC; Dan Nathan is short XLU June 38/36 put spread; Dan Nathan is long XLF short July 19 puts; Dan Nathan is long LULU July 65/70/75 call fly; Dan Nathan is long INTC long July 24 puts; Dan Nathan is long FB June/July 25 call spread;