How can client retrieve SQL Server public SSL certificate?
Technically it is possible, both TDS and TLS/SSL are documented. But you are asking the wrong question.

There is no point in willy-nilly manually installing a self-signed certificate everywhere, it buys you nothing. You will only install the MitM's cert everywhere...

Focus on deploying a properly configured PKI infrastructure. Either use a trusted certificate (ie. Verisign et all), or install a PKI trust in your network and use a infrastructure signed certificate.

See this series: Designing and Implementing a PKI