For the past two decades, the interaction of ultra-intense lasers with nano-foils has been motivated by acceleration of proton/ions for radiobiological applications. Despite some progress, the realization of a stable high-charge narrow-energy-spread protons of hundreds MeV remains a challenge. One promising scheme is the “light sail” acceleration, where laser pressure directly pushes the whole target, providing a strong accelerating force. One major drawback of such scheme is that it will suffer significant transverse instabilities that can break up the target, but the underlying mechanism has still not been clarified. Here we present a theoretical model that clarifies the origin of this long-standing problem, and support it with 2/3D PIC simulations for a wide range of parameters. Based on this understanding, we propose a new concept of relativistically intense laser tweezer for stable ion acceleration. In this scheme, two counter-propagating lasers of different colors collide on a nano-foil. By dragging plasma electrons out of the foil, such a tweezer simply avoids the electron-ion coupled instability and at the same time produces a strong micro-capacitor with a nearly uniform strong electrostatic field. This field preferentially accelerates protons with a narrow energy spread from a hydrocarbon target, indicating a new route for future compact laser driven ion accelerator.