No - Karkaroff had to be found to be killed.
The Dark Lord couldn’t use the Dark Mark to kill Death Eaters - when he wanted to kill Karkaroff for deserting, they had to actually track down where he was hiding to kill him.
“And they’ve found Igor Karkaroff’s body in a shack up north. The Dark Mark had been set over it – well, frankly, I’m surprised he stayed alive for even a year after deserting the Death Eaters; Sirius’s brother Regulus only managed a few days as far as I can remember.”
- Harry Potter and the Half-Blood Prince, Chapter 6 (Draco’s Detour)
If he could kill Death Eaters (individually) through the Dark Mark, then there’d be no reason to track Karkaroff down before killing him for his betrayal, since Karkaroff has a Dark Mark.
“When he touched the Mark of any Death Eater, we were to Disapparate, and Apparate, instantly, at his side. This Mark has been growing clearer all year. Karkaroff’s, too. Why do you think Karkaroff fled tonight? We both felt the Mark burn.”
- Harry Potter and the Goblet of Fire, Chapter 36 (The Parting of the Ways)