"I wouldn't necessarily say the cause for freer trade is over," Kudlow said Tuesday on CNBC's "Fast Money." "But we'll see who is appointed. I personally regret this [resignation] very much."
Kudlow said he's spoken with Cohn in recent days and "personally urged him to stay" in his post as President Donald Trump's economic advisor.
Cohn, a former Goldman Sachs president and free trade advocate, announced Tuesday evening that he would leave his post at the White House. The departure date has not yet been set.
"I'm quite sorry that this is true," said Kudlow. He said Cohn's resignation was "not great" for trade.
The news comes as Trump continues to move forward on his proposed steel and aluminum imports, pushing for a 25 percent tariff on steel and a 10 percent tariff on aluminum. Cohn and Trump's advisors did not agree on several important tariff issues.
Markets immediately dropped about 1.5 percent after the announcement of Cohn's resignation.
Kudlow, who called Cohn a "powerful force" in Washington, said Trump "tends to have balance" in appointing new people.
"I really would suggest to investors: Wait and see. Don't make any hasty judgments," he said. "Even when I worked with [former President] Reagan, we had staff changes at the highest levels, and life went on," he said. "Just don't panic over this."
The details of the proposed tariffs, including which countries or products would be affected, are still unclear. But last week's news of the tariffs immediately sparked fears of potential trade wars among investors.
Kudlow said blanket tariffs, or tariffs that include all countries, are the problem.
A better plan, he said, was to use targeted tariffs where there are problems, such as overcapacity.
"And you don't have to look far on a map," Kudlow said. "It's China. To me, targeted tariffs — that might lead to negotiations. And I would have gone after China. I wouldn't have gone after Canada. I think that is still an error. It's going to happen, I guess."
"There are other people in the White House who share this view: that trade actions would be okay as long as they're targeted and presumably temporary in lieu of negotiations," he said.