Your body’s resistance is roughly equal regardless of the voltage across it. Current goes according to Ohm’s law $V=IR$, so more voltage at fixed resistance means more current. More current is generally more dangerous, especially if it’s passing through you.

Your original reasoning incorporates a fallacy that power is constant in this situation.

Your body’s resistance is roughly equal regardless of the voltage across it. (See note at the end.) Current goes according to Ohm’s law $V=IR$, so more voltage at fixed resistance means more current. More current is generally more dangerous, especially if it’s passing through you.

Your original reasoning incorporates a fallacy that power is constant in this situation.

Note about resistance: To be more precise about the body's resistance, the skin has a much higher resistance than the rest of the body, and the skin's resistance, as noted in comments does vary based both on its condition and the voltage. There's some detail on that from from NIH. These variations, however, are less important at higher voltages and beside the main point above, which is that higher voltage still means higher current, especially at voltage levels at that are actually dangerous. I assume you weren't living in fear of the 9 V battery in your kids' toys.