Neymar scored his second goal of the game from the spot to put Brazil 2-1 ahead, and they went on to win 3-1. (Agencies)
Kovac was left fuming by the decision. "This was ridiculous today, and if we continue in this way we will have a circus," he told reporters at a post-match press conference dominated by questions about the penalty.
"If that's a penalty, we don't need to play football anymore. Let's play basketball instead. The lads played their hearts out but that was outright thuggery by a referee who was just out of his depth for a game of this magnitude."
Kovac, who said his team deserved at least a draw from a match played at a packed Corinthians arena in Sao Paulo, also felt Oscar's third goal near the end of the game should not have been allowed due to a foul earlier in the move.
"Was there a foul on my player? Yes. It was a foul. But that's life. We will march on. We will not be swayed."
Kovac, stony-faced yet calm, praised his team, which took the lead in the 11th minute after an own goal by Marcelo, and had several scoring chances during the rest of the match.
"My lads have fought for two years to reach the World Cup. They worked hard, prepared, and then they were faced with such a penalty decision here. You can imagine walking into our dressing room - what kind of mood my lads are in."
Nishimura ruled that Croatia's Dejan Lovren had fouled Fred, who appeared to throw himself to the ground, but Kovac did not blame the Brazilian for the incident.
"The referee was well-placed, he was not unsighted, he saw everything and he took that decision. I don't blame Fred at all."
Despite the setback, Kovac said he remained confident his team could progress out of the group. In their other two Group A games, Croatia play Cameroon in the hot and humid Amazon city of Manaus on June 18 and Mexico in Recife five days later.
"I am optimistic," he told Croatian television. "Naturally the lads are a bit down after this, but we will be up again as soon as we've had something to eat. We just have to make our way into the knockout stages because we are definitely good enough."
Kovac said he looked forward to welcoming back Mario Mandzukic into the team for the next two games, after he was suspended for the opening encounter.
Neymar scored his second goal of the game from the spot to put Brazil 2-1 ahead, and they went on to win 3-1.