"This is one of those cases that the rules were applied 100% correct as they're written today, for the rules of golf, but it doesn't mean that everybody doesn't feel horrible the next day," Whan told CNN on Monday.
"I feel horrible for Lexi Thompson, who proved she's one of the best golfers on the planet and certainly played well enough to earn a trophy," he continued.
"I feel bad for So Yeon Ryu who won a major but really didn't get the experience that most would get in winning a major, and I feel for the rules officials that had to do their job. They followed the rules as written."
"I cannot and I will not overrule a ruling of the game that was accurately portrayed," Whan added.
At Sunday's ANA Inspiration, the first major of the year in women's golf, Thompson was closing in on a second major title.
But the 22-year-old American was hit by a four-stroke penalty
in the final round after a viewer alerted officials to a rules infringement from Saturday's third round.
The penalties were for incorrectly replacing her marked ball on the 17th green in the third round on Saturday and signing an incorrect scorecard after that round.
Thompson, who was leading by two shots with six holes to play, was informed of the decision heading to the 13th tee. Thompson rallied but ultimately fell short in a sudden-death playoff to South Korea's Ryu.
After the conclusion of the ANA Inspiration, Whan said he tried to call Thompson but that it went to voicemail.
"I actually sat down late last night and wrote Lexi a letter that I mailed today," Whan said.
"I realized I didn't want to be the 450th voicemail or text message on her phone as she was going through this, changing planes. I wanted to make sure I got my point, my thoughts down on a piece of paper and sent to her."
Whan didn't say if the letter was an apology. "It was a really just a letter to say that sometimes the rules of the game, they feel unfair. I get that too.
"I can't overrule a rule that was accurately implemented by the rules of the game. There's nothing about this ruling that was wrong. In fact, it was 100% right.
"That's what makes it even more horrific, if you will, on the outcome," Whan continued. "I just wrote her a note to say I know this hurts. I'm incredibly proud of not only the way you handled it, but the way you played."
The governing bodies of golf, the USGA and the R&A, are considering implementing several rule changes
, though those changes wouldn't take effect until 2019 at the earliest.
"We'll sit down and talk about whether or not we want to separate from the rules of the game as they're written today," Whan said. "We haven't done that in our 67-year history. We've essentially followed the rules as written."
Ahead of the start of the Masters, which is the first major of the calendar year for the men, other professional golfers said they would support a rule change.
"I think it's unfair," Jimmy Walker said Monday in Augusta, Georgia. "There's no other sport where anybody can call in and say, 'Oh, that was a foul.' It just doesn't happen.
"I don't know why we're the exception, why you get to do that. No one gets to call the ins and outs in tennis. It just doesn't happen, so I think we need to change it."
"There's no question it should be ended," Rickie Fowler said. "I don't think you can find one player that would say otherwise."