South Korea's KOSPI fell 1.3 percent in early trade, taking its losses this week to 2.8 percent.
The Korean won also continued to skid, sliding 0.3 percent to 1,145 won to the dollar, after earlier sinking to its lowest level in a month.
MSCI's broadest index of Asia-Pacific shares outside Japan dropped 0.5 percent in its third session of declines, with Australia down 1.4 percent. Japanese markets were closed for a holiday.
Overnight, Wall Street closed sharply lower after Trump, with fiery rhetoric, warned Pyongyang against attacking Guam or U.S. allies after it disclosed plans to fire missiles over Japan to land near the U.S. Pacific territory.
Trump took specific aim at North Korean leader Kim Jong Un, saying, he had "disrespected our country greatly," and would not be "getting away with it."
The MSCI World index dropped 1.1 percent overnight in its third straight day of declines and its biggest one-day slide since May 17, as U.S. President Donald Trump stepped up his rhetoric against North Korea.
U.S. stock futures were marginally softer on Friday.
"The latest threats over North Korea have finally escalated to the point where market has been obliged to react," Ric Spooner, chief market analyst at CMC Markets, wrote in a note.
"U.S. markets had previously been becalmed amidst the Goldilocks scenario of strong profit growth, low interest rates and full valuations. Difficult to assess political risk is now intruding on this scenario."
The dollar extended losses against the yen to hit a new two-month low. It was down 0.1 percent at 109.07 yen, after retreating 0.8 percent on Thursday.
Japan is the world's biggest creditor country and there is an assumption that investors there will repatriate funds in a crisis.
Weakness in U.S. Treasury yields may also be supporting the yen, some analysts said.
U.S. Treasury yields fell to as low as 2.197 percent, their lowest level since June 28 overnight. They were at 2.201 percent early on Friday.
The dollar eased slightly against a basket of major currencies.
Gold prices, which hit a two-month low on Thursday, were steady at $1,286.16 an ounce, after surging over 2 percent in the past two sessions.
U.S. crude oil crude futures edged up 10 cents to $48.69 per barrel.
© Thomson Reuters 2017
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