Exhibit 23.1

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CONSENT OF INDEPENDENT REGISTERED PUBLIC ACCOUNTING FIRM

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We hereby consent to the incorporation by reference in the Registration Statements on Form S-1 (No. 333-284266 and 333-284263) and Form S-8 (No. 333-282544) of Adagio Medical Holdings, Inc. of our report dated March 27, 2025, which includes an explanatory paragraph regarding Adagio Medical Holdings, Inc.’s ability to continue as a going concern, relating to the consolidated financial statements of Adagio Medical Holdings, Inc. as of and for the years ended December 31, 2024 and 2023, which appears in this Form 10-K.

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/s/ WithumSmith+Brown, PC

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Whippany, New Jersey

March 27, 2025