“Our thoughts are with his family. I don't like violence, and it is very hard to understand when these things happen. Let's hope these things don't happen again in a football stadium,” said Zidane of the incidents which took place prior to Thursday evening's match.
Nine people were arrested, including three Russians and a Pole, during clashes outside San Mames in which a police officer died of a heart attack.
The incident also raised fears over hooliganism at this summer's World Cup which will be hosted by Russia.
“Football was invented to be enjoyed, for people to experience happiness or suffer with their teams,” said Barcelona coach Valverde, formerly in charge in Bilbao.
“But others use football as an excuse for other things, because football is seen by so many. It sullies football but it is not a problem for football, it is a wider problem. And of course we absolutely condemn that.”
FIFA said it “deeply regrets” the clashes in Bilbao but had “complete trust” in Russia's ability to ensure a safe World Cup — despite this being just the latest incident of hooligan violence involving Russians.
A minute's silence will be held before all games in Spain's top two divisions this weekend in memory of the dead policeman.
“La Liga reiterates its strongest condemnation of the violence and continues to work to eradicate it from football,” La Liga said in a statement.
Spanish Prime Minister Mariano Rajoy earlier offered his condolences for the death of the 51-year-old Basque police officer as he also “strongly condemned” the violence in the northern Spanish city.
The officer was one of over 500 deployed to San Mames for the high-risk Europa League last-32 tie.
He “suddenly felt ill during the serious incidents… despite being taken quickly to hospital, he suffered cardiac arrest,” the Basque government's security department said in a statement, adding his death was not “directly” linked to an injury he suffered in the clashes.
The trouble began as Spartak 'ultras' arrived at the stadium.
Some lit and threw smoke bombs and bins towards Athletic supporters, before police charged them.
Police detained five Spaniards, three Russians and a Polish man over the clashes, a spokesperson for the Basque security department said. They face charges of causing public disorder, rioting and disrespect for authority.
Five other police officers were injured, none of them seriously, and treated on the scene, the spokesperson added.
There was concern ahead of the match because of the reported presence of hardcore Russian fans in Bilbao, and police seized knives, bags full of rocks and brass knuckles from supporters of the two teams, the Basque security department said.
Earlier Thursday, German police arrested a Russian suspected of the “attempted murder” of a British football fan during the European Championship in France two years ago. He was detained at Munich airport as he was trying to travel to Bilbao to attend the match, according to German police.
But a spokesman for Russia's embassy in Germany, Denis Mikerine, “strongly protested” the arrest on Facebook, saying it was a “possible pretext” to “politicise the theme of football hooliganism before the 2018 World Cup in Russia”.
Russian authorities will be on high alert this summer for the World Cup after Russian fans were involved in pitched battles with England supporters in Marseille during Euro 2016.
Spartak won Thursday's match 2-1, but Athletic went through 4-3 on aggregate.
They will face Marseille in the next round in March, but Spanish rivals Atletico Madrid have been drawn against another Russian club in Lokomotiv Moscow.