There is nothing analogous to check in go. Go has "atari" which has some similarities, and is usually answered, but not always. The major difference is that in go you can reply to an atari with a counter atari. If someone threatens a 10 point move, you can threaten a 20 point move.
The same thing applies in chess; if someone attacks a piece with a pawn you can interpose an attack on their queen before responding to the original attack.
However, when it comes to the king you can't reply to a check by attacking their king; they would take your king first and win the game.
Chess would be more go-like if it was played winner take all - whoever had the last piece standing would win. So the king would just be another fighting piece, and the game would be won by killing all your opponents pieces. This would reduce the drama somewhat since there would be no amazing checkmates. But there would still be lots of amazing move sequences.
The first stone played is your "king" and if you lose that stone, you lose the game. Openings would be almost the same, except all of the variations where you sacrifice one particular stone are forbidden. This is similar to chess - when the king isn't involved, sacrifice variations are ok, but when the king is playing they aren't.
This could actually be interesting. If you absolutely cannot sacrifice one of your stones in a corner, what josekis are still available to you?
The problem with this is that once the stone was safe, it would basically be invulnerable. The opening might feature some extremely dangerous josekis, but after 50 moves it would be rare for this stone to be in danger. This is unlike chess where usually the king gets fairly safe in the middle game, but towards the end both sides nearly always have their king defenses broken down pretty badly, and both sides are just barely hanging on.