President Donald Trump is facing calls for a show of strength toward North Korea after Pyongyang’s weekend launch of a ballistic missile, posing the first major challenge to his administration by a foreign leader and an awkward balancing act with China.

U.S. lawmakers called for military exercises with regional allies, a rapid deployment of regional missile defenses and tough new sanctions. A senior administration official said the White House is unlikely to use authority it has to go after Chinese companies that do business with North Korea’s weapons sector.

The launch was a potent and troubling reminder of Pyongyang’s nuclear weapons ambitions as well as the limits of U.S. options. It came days after what White House officials consider a promising conversation between Mr. Trump and President Xi Jinping of China, which could prove a crucial partner in dealing with the threat from North Korea, among other issues.

The missile landed in the Sea of Japan on Sunday in Asia while Mr. Trump was hosting Japanese Prime Minister Shinzo Abe at his Mar-a-Lago estate in Palm Beach, Fla., prompting the leaders to make an unscheduled, late-night joint appearance.

Mr. Abe called the launch “absolutely intolerable” and demanded North Korea “fully comply” with United Nations resolutions banning such activity. Mr. Trump made no mention of North Korea, South Korea or possible retaliatory plans, saying, “I just want everybody to understand and fully know that the United States of America stands behind Japan, its great ally, 100%.”

World leaders are watching how Mr. Trump responds to Pyongyang’s move, which could provide clues to how the new U.S. administration will react to a range of other foreign-policy challenges and hot spots around the world.

U.S. defense officials believe the launch was of a medium- or intermediate-range missile, according to the Pentagon.

Early Monday, North Korea’s official Korean Central News Agency called the launch a successful test of a “surface-to-surface medium long-range ballistic missile,” which it called the Pukguksong-2. The North said that leader Kim Jong Un personally guided the missile launch.

In the report, Mr. Kim said the launch proved the North had made advances in various rocket technologies, and said the missile could be “tipped with a nuclear warhead.”

A former administration official said the Obama White House left the Trump team a playbook of options, anticipating a provocative action was likely immediately following the presidential inauguration last month. That included possible multilateral and unilateral moves encompassing sanctions, United Nations action, statements with South Korea and Japan, and speeding up the deployment of missile defenses.

The senior Trump administration official said Sunday the White House is in the midst of a policy review regarding North Korea.

Sen. Cory Gardner (R., Colo.) urged Mr. Trump in a written statement to “immediately pursue a series of tough measures,” including tough sanctions, military exercises with U.S. allies in the region, and the accelerated deployment in South Korea of an advanced missile system known as the Terminal High Altitude Area Defense system, or Thaad.

Other lawmakers echoed those prescriptions.

“It’s time the regime had to face consequences for its behavior,” Sen. Tom Cotton (R., Ark.) said in a statement.

A bill passed last year and signed into law by former President Barack Obama allows the administration to go after Chinese companies that do business with North Korea’s weapons complex. The law in effect forces companies to cut their ties to Pyongyang or suffer punitive financial consequences, such as restricted access to banking systems and financial networks.

The Obama administration used its provisions on at least one occasion last year, but lawmakers have urged a broader application, given that U.S. officials believe 90% of North Korea’s trade is with China, making Chinese companies an integral player in the country’s struggling economy.

“I think we have to tell the Chinese that they have to put the wood to North Korea in a much more serious way than they have done so far,” Senate Democratic Leader Chuck Schumer of New York said Sunday in an interview on CBS’ Face the Nation.

North Korea launched the missile off its east coast at 7:55 a.m. Seoul time on Sunday, in what officials in Japan and South Korea saw as both a provocation and a test of U.S. and Japanese responses.

South Korea’s Joint Chiefs of Staff said the missile flew about 500 kilometers (310 miles) before landing in waters between Japan and the Korean Peninsula.

“We suspect North Korea demonstrated a show of force in order to test the Trump administration and U.S. responses,” said a South Korean military official.

The U.S. Defense Department’s U.S. Strategic Command systems detected and tracked the launch of a medium- or intermediate-range ballistic missile at about 6 p.m. EST, according to a written statement from U.S. Strategic Command. The missile was launched near the northwestern city of Kusong and was tracked over North Korea and into the Sea of Japan, where it landed, the statement said.

South Korea’s acting President Hwang Kyo-Ahn said he and the government would push for a strong international response to punish North Korea, according to a written statement from his office.

Kim Kwan-jin, South Korea’s director of national security, convened a meeting of the Standing Committee of the National Security Council in response, a spokesman for the South Korean president’s office said. Mr. Kim also spoke by phone on Sunday with Michael Flynn, Mr. Trump’s national security adviser, the South Korean president’s office said.

Experts initially said the missile appeared to be one of North Korea’s so-called Musudan intermediate-range missiles, which are capable of reaching U.S. bases in Japan and Guam. The North has attempted to test the Musudan missile from the same launch site in the past, failing several times last year, though it claimed a successful test launch of the Musudan from a mobile launcher in June last year.

In a new-year address last month, North Korea’s Kim Jong Un said that the country was close to test launching an intercontinental ballistic missile, which would bring the North closer to being able to send a nuclear warhead to the continental U.S.

“It won’t happen!” Mr. Trump wrote a day later on his Twitter account, taking a tough stance toward North Korea’s nuclear ambitions before his inauguration. Earlier this month, Mr. Trump sent Defense Secretary Jim Mattis to Seoul and Tokyo, where he promised an “effective and overwhelming” response to any use of nuclear weapons against America or its allies.