Surface damage and material failure mechanism of CaF2 single crystal by 355 nm/6.8 ns laser pulses are systematically studied in this work. Surface damage characteristics, such as damage threshold, damage growth threshold, damage growth laws and damage growth coefficient of CaF2 crystal are carefully studied. Surface material cracking and local material vitrification (from CaF2 crystal to defective CaF2-x materials) are revealed. Material decomposition, oxidation and formation of Ca-0, Ca2+ and (F--F-) centers in surface layer are illustrated. Laser-driven thermal accumulation and thermal-stress gradient in CaF2 crystal are calculated and discussed based on the theory of Fourier heat conduction and thermal-elastic constitutive equations. Theory analysis suggests that the material failure process in CaF2 crystal is mainly driven by significant thermal stress gradient. Thermal gradient induce the material decomposition and local vitrification in CaF2 crystal. Combined results on surface damage and material failure behavior are beneficial to understand nanosecond laser-driven modifications and damage in fluoride crystals at ultraviolet wavelength.