Despite the "Socket" in the name WebSockets, and despite the fact that WS runs on top of TCP (with a HTTP-based initial handshake), WS is not TCP.
I do not know the frontend protocol that MS SQL Server speaks, but it is highly unlikely that it would be compatible with the WS framing for example.
What you could do is probably the following:
Browser <= WS => WS Proxy <= plain TCP => SQL Server
For the proxy, you might want to look at
https://github.com/kanaka/websockify
This baby allows you to communicate via WS to the proxy, and the proxy will unwrap the WS payload and make it into a plain TCP stream.
That way it should be possible to speak to SQL Server .. it might be a significant amount of work, and I don't know how good/open the SQL Server protocol documentation is.
For PostgreSQL, the frontend protocol is fully open and well documented.
Should it be unclear what I mean with above, I can go into more details .. or ping kanaka to ask what he thinks .. kanaka = author of the proxy and very active on WS anyway.