This term is mostly used when referring to business enterprises. If something is financially viable it means that it has a higher than not likelihood of success from an economic or business perspective (i.e. they have the funds necessary to succeed or they have a business model likely to succeed). I've most commonly heard the term used when referring to something's future performance, as in: "His proposal is not financially viable. I predict that his business will fail within the first year." Hope this helps!

"financially sound" perhaps.

Financial health? Casual speech wouldn't use "financial viability". We'd say the company is doing well. Or has a good business plan. Context for how you're wanting to use it might help.