The exercises, which also have involved US B-1 bombers, are creating "the worst ever situation prevailing in and around the Korean Peninsula", North Korea's UN Ambassador Ja Song Nam said in a letter to the secretary-general on Monday.
The soldier, whose rank and identity have not been disclosed, was flown by helicopter to hospital on Monday after his escape to South Korea, during which he was sacked on by four North Korean soldiers.
The soldier was being operated upon for a second time on Wednesday, the hospital treating him said.
About 30,000 North Koreans have defected to South Korea since the end of the 1950-53 Korean War, but most travel through China.
"There will need to be some questioning on why he defected after his treatment is over", ministry spokesman Baik Tae-hyun told a regular briefing.
Hospital officials where under strict security agency orders not to talk to media and all updates on the soldier had to be through the military, workers there told Reuters.
The United States has said that all options, including military, are on the table to deal with North Korea, although its preference is for a diplomatic solution.
The military drills come amid U.S. President Donald Trump's visit to Asia, which has been dominated by discussions over the North Korean nuclear threat.
American presidents often visit Panmunjom and other DMZ areas during their trips to South Korea to reaffirm their security commitment to the South. Trump planned to visit the DMZ to underscore his stance against North Korea's nuclear program when he came to South Korea last week as part of an Asian tour, but his plans were thwarted by heavy fog that prevented his helicopter from landing at the border area.
The solider was found beneath a pile of leaves on the southern side of the JSA and South Korean troops crawled there to recover him.
North Korean soldiers have occasionally defected to South Korea across the border.
It's unclear what the soldier's conditions are or why he chose to defect to the South.
Panmunjom, once an obscure farming village inside the 2 1/2-mile-wide Demilitarized Zone that separates the two Koreas, is where an armistice was signed to pause the Korean War.
The Joint Security Area was the site of some bloodshed during the Cold War but there hasn't been major violence there in recent years.
It said an investigation into Monday's event was ongoing.
Two American soldiers also were killed in the DMZ by ax-wielding North Korean soldiers in a 1976 brawl over an attempt to trim a poplar tree.
North Korea has typically accused South Korea of enticing its citizens to defect, something the South denies.
The defection came during escalated tensions between the North and the South and almost a week after President Trump visited Seoul for his five-country Asia tour.