The U.S. imposed new 25 percent tariffs on steel and 10 percent tariffs on aluminum, imported from the European Union, Canada and Mexico.

So, with the confidence of a strong economy and rising stock market, "[Trump's] got his foot on the accelerator," Clifton said.

Trump "got elected and campaigned on this, and he's tired of being told by his advisors what he can do," Clifton added.

"I think more and more of the cabinet are becoming pro-trade, but I think the president is becoming more bombastic on trade because he believes the U.S. got a raw deal for decades," said Daniel Clifton, head of policy research at Strategas.

Already, G-7 finance ministers, meeting in British Columbia over the weekend, chastised the U.S. for the trade skirmishes brewing across the world. Last week, Japan called for rational debate to prevent protectionism from harming the global economy.

The heads of G-7 nations travel to Quebec at the end of the week for a meeting hosted by Canadian Prime Minister Justin Trudeau and including leaders of Germany, Italy, France, UK, and Japan, all countries subject to U.S. tariffs.

There could be major developments on the trade front with China during the coming week, as well, and more clarity on whether revisions to the North American Free Trade Agreement can be concluded this year.

With a lack of earnings and major economic reports, markets could focus more heavily on trade-related headlines in the coming week, and they could get nasty as President Donald Trump prepares to meet world leaders in Canada.

The tariffs prompted swift retaliation and raised doubts about whether the administration will be able to reach a near-term agreement with Canada and Mexico on a revamped North American Free Trade Agreement. But many market strategists say those tariffs alone will not rock the stock market or economy unless they escalate.

The bigger question is whether the U.S. can resolve trade disputes with China and come to a new NAFTA accord. As for China, Commerce Secretary Wilbur Ross touched down in Beijing Saturday, amid a widening disagreement over whether China would agree to commit to long-term contracts on U.S. energy and agricultural products.

Clifton said there could be positive developments on China trade, but the outcome for NAFTA is unclear. "It could turn out we get a good China negotiation next week," he said, noting the disagreement over contracts could be posturing.

"We will not have a better determination of what's happening in NAFTA until we see what goes on in China," he added. "As an investor, what you should be worried about is NAFTA going away. That is a meaningful change to the supply chain. That is different than the tit-for-tat tariffs on goods."

The Trump administration announced the metals tariffs after Mexico and Canada would not agree to a sunset clause on the free trade accord, arguing it would be difficult to make decisions if the agreement is not permanent. The demand was made as the leaders of Canada and Mexico were discussing coming to the U.S. to close the deal.

"The U.S. has to think of its leadership role and right now the direction we're leaning is not very constructive," said Tom Block, Fundstrat's Washington policy strategist. "I do think the economy right now is running on every cylinder. It's so strong. ... If you're quarter-to-quarter focused, the next few quarters look pretty good. But if you start to take a deeper look and say how sustainable is this, that's when the trade and tariffs are a problem."

Strategists said the administration is coming down to the wire to reach a NAFTA deal and is just days away from a deadline under which it can still receive approval from the current Congress.

"Everybody would be caught by surprise if something comes out in the next week," said Juan Carlos Hartasanchez, Albright Stonebridge Group senior director.

"What's important is Mexico and Canada did not budge to the will of the U.S. ... That in itself reduces the tension to reach an agreement in the short term. Now the real-time horizon to reach an agreement for NAFTA has expanded to what we initially thought — more toward 2019," he said.

In the week ahead, there is international trade data coming out on Wednesday, as well as productivity and costs data, which includes a look at inflation. The PCE data is the preferred inflation measure watched by the Fed. There are no Fed speakers in the coming week since it is the quiet period ahead of the June 12 and 13 Federal Open Market Committee meeting, when the Fed is expected to raise interest rates.

"You can [worry] tariffs are going to be bad for the economy, but it's juxtaposed against strong numbers that we've seen in today's unemployment report. We don't have that [buffer of the data] next week. We could revert to what could go wrong, and we will hear more about retaliation," said Art Hogan, chief market strategist at B. Riley FBR.

So far, Europe has pledged its own tariffs on products such as Bourbon and Harley-Davidson motorcycles. Mexico said it may attach tariffs to cheese, apples, rolled steel and pork, while Canada said it would put tariffs on steel and aluminum and other products.