We report on a noninvasive and noncontact all-optical method to measure the elasticity of the cornea. We use a pulsed laser to excite surface acoustic waves (SAW) that propagate on the corneal surface, then use a phase-sensitive optical coherence tomography system to remotely record the SAWs from which the corneal elasticity is estimated. In addition, the system is able to provide real-time tomographic images of the cornea being examined, an important consideration for clinical studies. While precisely maintaining a range of intraocular pressures (IOP), a series of measurements is performed on ex vivo intact primate eyes. The measurement results not only demonstrate the feasibility of the proposed system to remotely measure the corneal elasticity, but also suggest a strong correlation between the corneal stiffness and the true IOP.