Technically yes, the terms under which a Schengen visa is issued require you to have health insurance up to a certain amount of coverage. You may be asked for proof of this in the form of health insurance documents when you're entering the Schengen zone and/or at any passport control checks.
In practice though, it is unlikely that you will be asked to provide documentation for this. However, travel health insurance is still a good thing to have as hospitalisation costs in case something does happen can be expensive. You don't need to get an expensive policy; at least in the UK, you can get short term travel health policies which cost less than £10. Even though these policies may have high deductibles, for serious accidents they will still help you recover medical expenses.