Inter president Massimo Moratti gave coach Rafael Benitez his backing after last weekend's derby defeat by leaders AC Milan, but another loss has left the Spaniard exposed with the champions now nine points off top spot.
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Eto'o was not cautioned by the referee for a headbutt but the Cameroon striker could be punished retrospectively.
He scored Inter's consolation goal late on after Sergio Pellissier and Davide Moscardelli put Chievo two up.
"It's clear after a defeat no one is happy but the only thing I can say is the players worked hard and the reaction in the second half on an awful pitch was the reaction we expect," a solemn Benitez, whose side have taken two points from four games, told Sky television.
"What Eto'o did is not something good but he took a punch first and it was provocation. He lost a bit of control, that's clear."
With five sides above Inter after 13 games, last season's treble winners have a fight on their hands to win a sixth successive scudetto given their poor form and injuries.
Pellissier's first-half header and Moscardelli's second-half strike secured Chievo's victory, but the game will be remembered more for the actions of the league's top scorer, Eto'o.
In a moment of madness similar to France midfielder Zinedine Zidane's assault on Italian Marco Materazzi in the 2006 World Cup final, Eto'o headbutted Cesar in the chest after being hit by the defender earlier.
Eto'o netted his ninth goal of the Serie A season in stoppage-time on a rain-sodden Verona pitch, where Italy's rugby side had played last weekend.
Inter had three youth team players on the bench because of their injury problems, which Benitez has said are bad luck and nothing to do with his changing the team's training regime after taking over from ex-coach Jose Mourinho.
