The question of whether quantum tunneling imposes a finite time delay is a fundamental and contested issue, prominently debated in strong-field atomic ionization. Here we extend this debate to the molecular domain by investigating the tunneling dissociation time of the proton in ${\mathrm{H}}_2{}^{+}$. Through numerical solution of a two-level time-dependent Schrödinger equation and analysis via the backpropagation method, we discover a significant and tunable delay in the proton's emergence. Crucially, we demonstrate that this delay is not intrinsic but is primarily governed by the relative timescales of two motions. The delay's sign is determined by the competition between the laser cycle and the molecular rotational period: It is positive when the laser cycle is shorter than half the rotational period and negative in the opposite regime. Our work underlines the importance of tunneling dissociation delay by revealing its physical origin, establishing laser-driven molecular rotation as a powerful control knob and tunneling delay as a real-time probe of ultrafast rotational dynamics.

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• Molecular dissociation
• Multiphoton or tunneling ionization & excitation
• Ultrafast phenomena
• Ionized molecules
• Dipole approximation
• Schroedinger equation
• Semiclassical methods
• Two-level models