Chemistry - Do acids really donate a proton?

Yes, according to the Arrhenius theory, acids dissociate in aqueous solution and release a proton ($\ce{H+}$). The Brønsted–Lowry defines acids ($\ce{HA}$) and bases ($\ce{B}$) in such a way that their interaction is characterized by the exchange of a proton according to $\ce{HA + B <=> A- + HB}$.

However, this is about the reaction of molecules in which real hydrogen atoms are bound to other atoms.

To illustrate the idea, let's think about the reaction between acetic acid, $\ce{CH3COOH}$, and benzylamine, $\ce{C6H5CH2NH2}$. If we combine both liquids in our imagination, the following will happen:

$\ce{CH3COOH + C6H5CH2NH2 -> CH3COO- + C6H5CH2NH3+}$

The acid donates a proton to the amine, which serves as the base. In the course of this reaction, the acid is converted to its corresponding base, while the base is converted to its corrsponding acid.

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You are confusing dissociation and proton exchange with a type of radiocative decay known as proton emission.