Yes the phone would work on Verizon, but the 4G wouldn't. Verizon uses LTE (Long Term Evolution) for its 4G. The earlier Sprint 4G phones use a different 4G type called WiMax which is not compatible with LTE. Even though most of the carriers are rolling out LTE, their phones won't really work on each others' network due to frequency differences in where the companies run their 4G.
In a nutshell: Your phone's voice and 3g should work on Verizon. The 4G won't. If 4G is important then you may want to consider the outright purchase of another phone so you won't have a contract.
FYI-Only AT&T/T-Mobile require SIMS for their phones.