The Nikkei 225 fell by an unusually wide margin of 5 percent to 19,155.14. The Shanghai Composite Index ended off 0.9 percent at 2,504.82 after being down as much as 2.3 percent at midday. Benchmarks in Thailand and Taiwan also declined.
Markets in Europe, Hong Kong, Australia and South Korea were closed for Christmas.
Wall Street indexes fell more than 2 percent Monday after Mr. Trump tweeted that the Federal Reserve Board was the U.S. economy's "only problem." Efforts by Treasury Secretary Steven Mnuchin to calm investor fears only seemed to make matters worse.
U.S. stocks are track for their worst December since 1931, during the Great Depression.
Shanghai is down almost 25 percent this year. Tokyo, Hong Kong and other markets are on track to end 2018 down more than 10 percent.
The market has been roiled by concerns about a slowing global economy, the trade dispute with China and another interest rate increase by the Fed.
Mr. Trump's Monday morning tweet heightened fears about the economy being destabilized by a president who wants control over the Fed. Its board members are nominated by the president, but they make decisions independently of the White House. The board's chairman, Jerome Powell, was nominated by Mr. Trump last year.
"The only problem our economy has is the Fed," the president said on Twitter. "They don't have a feel for the Market, they don't understand necessary Trade Wars or Strong Dollars or even Democrat Shutdowns over Borders. The Fed is like a powerful golfer who can't score because he has no touch - he can't putt!"
Mr. Trump has made the stock market a key barometer of his success, pointing to its strong performance earlier this year as proof his policies are working. But as equities have slumped, he has repeatedly attacked the U.S. central bank for raising interest rates, calling it his "biggest threat" and describing it as "out of control."
More pointedly, Mr. Trump has publicly expressed regrets about appointing Jerome Powell as Fed chairman. Last week, Mr. Trump reportedly consulted advisers about firing Powell, Bloomberg reported.
"If Trump tries to remove Powell, it could throw the financial markets into deep chaos that makes the current market turmoil look tame," analysts at Oxford Economics said in a note.
The Standard & Poor's 500 index slid 2.7 percent Monday to 2,351.10. The benchmark index is now down 19.8 percent from its peak on Sept. 20, close to the 20 percent drop that would officially mean the end of the longest bull market for stocks in modern history - a run of nearly 10 years.
The Dow Jones Industrial Average sank 2.9 percent to 21,792.20. The Nasdaq skidded 2.2 percent to 6,192.92.
On Sunday, Mnuchin made a round of calls to the heads of the six largest U.S. banks, but the move only raised new concerns about the economy.
Most economists expect U.S. economic growth to slow in 2019, not slide into a full-blown recession. But the president has voiced his anger over the Fed's decision to raise its key short-term rate four times in 2018. The moves are intended to prevent the economy from overheating.
Technology stocks, health care companies and banks took some of the heaviest losses in Monday's sell-off. Wells Fargo slid 3.4 percent, Microsoft 4.2 percent and Johnson & Johnson 4.1 percent.
U.S. markets reopen Wednesday after being closed for Christmas.
In energy markets, Brent crude, used to price international oils, lost 9 cents to $50.68 per barrel in London. The contract plummeted $3.33 on Monday to close at $50.77.
In currency trading, the dollar gained to 110.15 yen from Monday's 110.45 yen. The euro advanced to $1.1417 from $1.1405.