Is it legal for a law firm to go public in the US?

It would be different in every state, since each state's laws are different and law firms are regulated by which ever state's bar association they fall under the jurisdiction of, but: the short answer is it would likely be a violation of the attorney's professional responsibility/ethics standards. In Florida it is against the bar's rules for a non-lawyer to have an ownership interest in a law firm that would compromise the attorney's professional judgment. This would preclude any type of public trading, since companies are responsible to shareholders before employees. A lawyer's professional judgment and the interests of the client must be put before the profit of the company. Similarly, a lawyer is generally (with few exceptions) prohibited from profit sharing with non-lawyers In Florida. Even more rules place further limitations on this type of situation. Most states probably have similar laws/rules. 

This is not a substitute for legal advice. Seek personal legal advice from a qualified lawyer licensed to practice in your jurisdiction. Nothing In this answer creates a lawyer-client relationship. I am a lawyer, not your lawyer.

Additionally, The NYT had an article a few years ago about this: Selling Pieces of Law Firms to Investors

Lawyers will claim fealty to their notions of ethics for why this is prohibited; nonetheless, it is a poor business decision on their part.

Allowing outside investors would accomplish two things:

• Provide for a more diverse source of capital than the current practice of having partners buy into their firms
• Allow partnerships to raise capital from banks or other institutions in the form of debt (note this is somewhat different than a revolving line of credit with one's bank)

Right now, I believe that the only way a law firm facing a liquidity crunch can raise money is by (1) requiring its partners to cough up more money or (2) factor receivables.  I suppose they could sell off furniture, too, if they were really desperate.

None of those options is really a viable source of capital long term.  Many of the law firms that have collapsed in recent years have collapsed precisely because they have no source of capital other than what has been paid in by partners.  It's a very leveraged (and risky) business model.