The Short Answer: No.
A King can never capture another King. They cannot even stand next to each other.
One of the most common questions from beginners is: "Can I kill the enemy King with my King?" The answer lies in the most fundamental rule of chess safety: Check.
In chess, it is illegal to move your King into "check" (danger). If you were to move your King to a square right next to the enemy King, your King would be under attack by that enemy King.
Because they can never get close enough to occupy the same square, they can never "eat" or "kill" one another.
No. Because a King can never get close enough to attack the enemy King without putting itself in danger first, a King cannot deliver a check.
Checks are delivered by:
The King is the target of the attack, not the attacker of the other King.
If you are playing a game and you accidentally move your King to a square adjacent to the opponent's King, you have made an illegal move.
If you have captured all your opponent's pieces, and they have captured all of yours, so only the two Kings remain on the board, the game is immediately a Draw.
Since neither King can checkmate the other, it is impossible to win. This is called a "Draw by Insufficient Material."
Now that you know the rules of the King, learn how to use him in the endgame or checkmate the opponent using other pieces: