Lopetegui had been unbeaten in his 20 matches in charge but was relieved of his duties just days before Spain's opening match against Portugal after agreeing to become the new Real Madrid boss following the tournament.
Fernando Hierro was named as stand-in coach and, although Spain finished top of Group B to reach the knockout stages, they crashed out at the last-16 stage due to losing on penalties to Russia on Sunday.
"I don't regret the decision taken with Lopetegui because it was done with conviction and values that aren't influenced by subsequent results," Rubiales is quoted by the Irish Times as saying.
"When a decision is made with responsibility and, above all, with values, one has to be calm. All my board of directors gave me unanimous support, we are very calm. Now we need to analyse things and see what's best for the national team and the federation.
"What I have clear is that I'm proud of the way Hierro and his staff have behaved. He has assumed this responsibility with bravery. In the coming weeks we will see what's best for everyone.
"You're always learning things, now it's very difficult to analyse everything in an hour. We're the toughest judges of ourselves and we'll see what things can be improved."
Spain won the World Cup in 2010 but were knocked out at the first hurdle in Brazil four years later.