The injury-plagued 27-year-old scored a Premier League hat-trick against Arsenal last week, prompting calls for Hodgson to include him in the squad for the tournament in France, which starts in June.
"Every time someone scores a goal and he's English it is 'why is he not playing for England?'," the 68-year-old told reporters.
"I'm certainly not going to pick him on the basis that he scored a hat-trick so therefore has to be in the England team."
Carroll's main strengths are his physical presence and ability in the air, offering a different threat to Hodgson's other forward options including Harry Kane, Jamie Vardy, Daniel Sturridge, Danny Welbeck and Wayne Rooney.
"We know Andy, we know his qualities," Hodgson said.
"In an ideal world, if you had a lot of places at your disposal, you'd always include a guy like him because he is exceptional in the air - but whether or not I'd be able to include him with all these others, I don't know."
Carroll won the last of his nine England caps in 2012.
(Reporting by Ed Osmond. Editing by Patrick Johnston)
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