Lee emphasized in his cabinet meeting that Ulchi Freedom Shield is a purely defensive exercise.
"A strong alliance makes the foundation of security stronger, and strengthening our own capabilities increases our value and necessity as an ally," Lee said.
He stated that South Korea's top priority is maintaining peace with North Korea, but said the country must be prepared for a "worst-case scenario."
Lee specifically mentioned boosting South Korea's nuclear-powered submarine program, among others.
Trump's statement on Monday, posted to Truth Social, was openly critical of Lee's administration with respect to North Korea.
"Based on my very good relationship with Kim Jong Un, of North Korea, I am not happy with the fact that the United States has, long ago, agreed to participate in Joint Military Exercises with South Korea," Trump wrote.
"These exercises are not only costly, with much of these costs paid for by the United States of America (as usual!), but send a signal that is totally inappropriate and hostile, to a Country that, as long as Donald J. Trump has been President, has been unthreatening and respectful."
Lee's administration noted the kind words for North Korea in his statement on Tuesday, with Lee's national security advisor, Wi Sung-lac, remarking that he hopes Trump's relationship with Kim could lead to lasting change.
"We hope the friendly relationship between the U.S. and North Korean leaders will lead to meaningful North Korea-U.S. dialogue," Wi said at a Tuesday briefing.
Trump also took the opportunity to highlight Lee's apparent refusal to assist the U.S. in the conflict with Iran.
"While somewhat unrelated (?), I recently asked the President of South Korea if they would like to join us in the Denuclearization of the Islamic Republic of Iran, and they said, ‘No thanks!’" Trump wrote.
Reuters contributed to this report.