No, the A1 certificate is not a health insurance certificate.
In addition to the issuance of an A1 certificate, you must have either:
the European Health Insurance Card, if the stay abroad is temporary,

or
an S1 certificate, if the residence is transferred to the country of employment.

The S1 certificate is issued by the Iceland Health (sjukra.is).
When applying for an A1 certificate, you can mark in the application that the Social Insurance Institution requests an S1 certificate from the Iceland Health on behalf of the applicant.