BBC Sport understands Rooney was surprised to have been left out of the starting XI as England drew 0-0 to finish second behind Wales in Group B.
The Manchester United player was one of six changes, with Hodgson criticised after his decision backfired.
"I never have regrets if the team has played well," said Hodgson.
Hodgson's future depends on how England fare at Euro 2016 and a runners-up spot means a potentially trickier route through the knockout stages.
France, Spain, Italy and Germany are all now on England's side of the draw, rather than that of Wales.
The 68-year-old's contract expires at the end of the tournament and Football Association chairman Greg Dyke said it will only be renewed if England "do well" - with reports that Monday's tactical gamble has undermined his chances of a new deal with an unimpressed FA hierarchy.
This was no time to take risks and yet Hodgson blindly walked into a trap of his own making
"I would have maybe had some regrets if we hadn't played well and dominated the game and controlled it," Hodgson added.
England created 65 chances but scored just three times in their three Group B matches, having drawn 1-1 with Russia and beaten Wales 2-1 before Monday's draw in Saint-Etienne.
They face the second team in Group F in Nice on Monday (20:00 BST) - which could be Portugal, Austria, Hungary or Iceland. They play their final group matches on Wednesday.
England scouts, rather than Hodgson himself, will watch Iceland v Austria in Paris, travelling from England's nearby Chantilly base.
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The former Fulham, Switzerland and Inter Milan coach said the only "fault" was not taking chances.
Harry Kane, Jamie Vardy, Daniel Sturridge and Marcus Rashford are the four recognised strikers in England's squad, while Raheem Sterling and Adam Lallana have featured in attacking wide positions and record scorer Rooney has been played in midfield.
Of his striking options Hodgson said: "All of them have shown a lot of qualities in the games but none of them, as yet, have really shown they are able to score the goals we need."
England manager Roy Hodgson says he has no regrets over his team selection against Slovakia despite captain Wayne Rooney's surprise at being dropped.