No.

Obviously, Prideaux does not kill them directly (e.g. by shooting them). That said, we might still consider him responsible for killing them if his actions lead directly to their deaths.

But for that to be true, they actually have to be dead, and I doubt that Karla would kill his prisoners simply because Haydon is dead. It's been a long time since I saw the BBC version, but I don't recall there being any actual evidence that Karla kills them, and there is no such evidence in the 2011 Hollywood version.

Why would he not?

Karla has other agents, some of whom are doubtless already in British custody, and he'll still need collateral to barter for them even if Haydon is no longer available. And, as Roy notes in the second clip, "Moscow Center prides itself on getting its people back." So, bargaining for the return of agents is a longstanding agency policy, not likely to be abandoned simply because one agent was killed.

Killing all his prisoners in retaliation for Haydon's death would literally bankrupt him in terms of hostage-exchange, and even worse, it would likely end the current arrangement under which such exchanges are possible. A spymaster would only pursue a scorched-earth tactic like that if he believed he would never again need to get some of his own captured people back.

No, the likely consequence of Haydon's unsanctioned assassination by Prideaux is that the current negotiations will be disrupted and have to start over, and it's possible Karla will try to drive a harder bargain this time to exact some kind of price for Haydon's death, demanding more agents in return, or higher-value agents, etc. Maybe the Circus will even agree.

Finally, it's worth noting that Haydon is not actually a Russian agent, he's a Brit who defected. This means Moscow Center didn't pay for his training or his salary, etc. So, the loss of Haydon is less costly to them than with a typical agent. (Yes, Witchcraft is bust, but that would have been true even if Prideaux had not killed Haydon.)