If I signed a employment contract and was required to give 60 days notice but only gave 20 days. Can the employer sue me?

There are not enough facts here to give a meaningful answer, but generally, yes, if you did not give the required notice, you breached the contract. Several issues arise, however, that might affect whether the employer actually sues you, including: do you have a non-compete built into the employment contract (industry/location specific)?; what is the basis for the 60-day notice (need to hire replacement)?; will the employer suffer any damages as a result?; how long was the contract, such that the shorter notice does not materially affect your obligations under the contract?; did any of your obligations survive termination (i.e., are you working on any long-term projects that need your input)?; are you leaving early for matters beyond your control (force majeure events, such as death in the family, medical issues, etc.)?; and finally, did the employer say it was okay for the shorter notice, or did he give you any indication he was "waiving" the requirement? Read the contract again carefully.