There's no hoopla at the end of The Seagull. After the unhurried pace of the rest of the play the ending—with just four lines after the gunshot—almost feels rushed. Konstantin has killed himself without fanfare (and done so offstage, thank goodness).
Chekhov seems to be making the point that even this extreme action, suicide, is just one of the things that happens in life. It shouldn't be spotlighted or given a musical number. Well, maybe one musical number.
It's also interesting that Chekhov cuts the play off before Arkadina discovers her son's death. She really should know what's happened—she even says "Oof! That scared me! It reminded me of when…" implying that the first suicide attempt is still on her mind (4.189).
But Dorn is easily able to pull a bait-and-switch on her, blaming the gunshot sound on an exploding bottle in his medicine bag (good one, Dorn). Is Arkadina in denial? Is she afraid of her own guilt in the situation? We don't know for sure. With the protagonist dead, Chekhov wraps up the story quickly and simply leaves us with questions about the future of the family who survives him.
We've said it before, and we'll say it again: thanks, Chekhov.