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.
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"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)
