Hi,
If you are only restarting IIS and not SQL Server, then this should not impact users that are connected directly to SQL Server since they are not accessing via a web application. Are SQL Server and IIS on the same server? If yes, you are not planning on restarting entire server, just IIS, correct?
In regards to your certificate issue, before doing anything I recommend you open IIS Manager and check the binding to make sure that the correct certificate is bound. It is entirely possible that you just need to correct 443 binding and your web site will work fine.
If the above was helpful please click Accept Answer.
Thanks.
-TP