The announcement came as Benjamin Netanyahu visited the White House in a trip the Israeli prime minister said he cut short after an early morning rocket, allegedly fired from the besieged Gaza Strip, struck a home in central Israel, wounding seven people.
Israel began striking Hamas targets in Gaza later on Monday, the Israeli military said. Hamas had earlier denied its movement was behind the overnight rocket.
Monday's decree formalised Trump's statement last week, saying it was time for the US "to fully recognise" Israeli sovereignty over the Golan. The move appeared to give Netanyahu a boost before the closely contested April 9 Israeli elections.
"This was a long time in the making," Trump said alongside Netanyahu in the White House.
"Today, aggressive action by Iran and terrorist groups in southern Syria, including Hezbollah, continue to make the Golan Heights a potential launching ground for attacks against Israel - very violent attacks," Trump said. "This should have been done numerous presidents ago."
Syria's foreign ministry called the US decision to recognise Israeli sovereignty over the Golan a "blatant attack on the sovereignty and territorial integrity" of Syria, according to a statement carried by state news agency SANA.
"The liberation of the Golan by all available means and its return to the Syrian motherland is an inalienable right," the statement added. "The decision ... makes the United States the main enemy of the Arabs."
Turkish Foreign Minister Mevlut Cavusoglu said it was impossible for Turkey to accept the US decision on the Golan Heights.
He added that action would be taken against the US over the move, including at the United Nations.
UN Secretary-General Antonio Guterres was "clear that the status of Golan has not changed," UN spokesman Stephane Dujarric said on Monday.
"The UN's policy on Golan is reflected in the relevant resolutions of the Security Council and that policy has not changed," Dujarric said.
A UN Security Council resolution adopted unanimously by the 15-member body in 1981 declared that Israel's "decision to impose its laws, jurisdiction and administration in the occupied Syrian Golan Heights is null and void and without international legal effect". It also demanded Israel rescind its decision.
Qatar's Ministry of Foreign Affairs also issued a statement rejecting the US move and saying the Golan Heights are occupied Arab land.
The Arab League condemned the move saying that "Trump's recognition does not change the area's status."
Earlier this month, a senior US administration official told reporters in Washington that there was "no change in [US] outlook or our policy vis-a-vis these territories and the need for a negotiated settlement there".
The official was responding to questions about why the US changed its description of the Golan Heights in its latest annual human rights report in which the area was referred to as "Israeli-controlled", not "Israeli-occupied" as it was previously stated.
Trump, who has shown robust support for his country's close ally Israel, recognised Jerusalem as the country's capital in 2017, defying international consensus and angering Palestinian leaders, who view Israeli-annexed East Jerusalem as the capital of their future state.