Not quite-
A polar protic solvent will actually hydrogen bond to the leaving group thus helping to distribute the negative charge that the leaving group carries. This, in turn, stabilizes the carbocation by preventing the leaving group from reforming a bond with the carbocation.
With an SN2 reaction, the nucleophile must approach the substrate from the backside of the leaving group. Since steric factors, rather than electronic factors as in SN1, are of paramount importance with SN2, anything that interferes with the nucleophile attacking will slow the reaction rate. Polar protic solvents, by definition, are capable of hydrogen bonding. If there is a polar protic solvent with a strong nucleophile, the solvent molecules hydrogen bond to the nucleophile forming a "solvation shell" which greatly increases the effective size of the nucleophile. This in turn prevents the nucleophile from attacking from the backside of the leaving group and greatly slows the reaction rate.