Limited Liability Laws - Without limited liability laws, the economy would not have access to the capital it needs to grow and prosper.
You can still limit responsibility of the investors, if the directors of the company take the blame for screwing up. E.g. If someone gave a business money, and the person running that business went and did something extremely immoral, I wouldn't hold the investor responsible, and don't need the state to tell me that. In short, Limited Liability doesn't need government, just society agreeing that it's fair this way.
Property Rights - Corporate property rights – one of the main legal instruments that insulate business from government power – can be created and maintained only by government.
Not quite sure how this is better that non-corporate, plain old business or personal property ownership, which doesn't need government, just enough people agreeing that you own what you say you own.
Law and Order - Without the rule of law, our economy would resemble the “mafia capitalism” that Russia has suffered from in its transition to capitalism.
Mafia still charges way less for protection than government (15% compared to 45%), and understands that going to war is bad for business. Russia suffers from mafia in the government, not from mafia. Besides, "Extortion, bribery, kidnapping, and murder," aka "taxation, bribery, arrests, and capital punishment (or war)," is something we have under government already. So the other system can't really be worse in this regard.
Bankruptcy Protection - Bankruptcy laws protected otherwise healthy businesses that were temporarily short of funds.
Why can't orderly, private agreements on debt restructuring between a debtor and borrowers exist without government? What does government enforce that private parties can't agree on on their own? Surely private investors understand that they can get more money from a business that ran into temporary trouble by allowing it to restructure their debt agreement, than from liquidation? And in some cases, investors that actually study the market can recognize a business that should be liquidated better than someone who is emotionally attached to it and in denial.
A Stable Money Supply - Widespread commerce and a stable economy both require a stable and dependable money system – one in which consumers and merchants have faith. This can only be provided and maintained by the federal government.
Well I guess we're proper fucked then (See "Bitcoin")
Patents and Copyrights - Bill Gates likes to think of himself as a self-made man, but he would not be one of the richest men in the world if the government did not make it illegal for anyone but Microsoft to copy and sell Windows.
And yet, Microsoft is making billions in China, where it has no copyright protections at all. It just charges a much smaller fee ($1 to $3 for a Windows CD), people buy it because they know their copy isn't full of viruses and is more convenient to get, and Microsoft makes their money on volume and providing follow-up service. And Microsoft isn't the only company doing this. Services like iTunes and Netflix are also surviving in the age of extremely easy music and movie downloads, by charging for the service instead of the copyrighted content. Plus you have trade secrets, first-mover advantages, expertise in a subject you invented, etc.
Banking Regulation and Insurance - Banks cannot survive runs because they have loaned out most of the money deposited with them and therefore cannot pay it out to a large number of depositors at once...government was there to guarantee those deposits
If there was no one to guarantee deposits, perhaps banks would revert to using savings for risky investments, and keeping checking deposits safely in the bank? Sounds like this is a problem with the banks that government let's them get away with. Regardless, see "Bitcoin."
Corporate Charters - Capitalism today is corporate capitalism. But the corporation itself is a creation of government. Corporations can come into being only through charters: the legal instruments by which state governments allow businesses to incorporate.
See reply to Limited Liability Law (no reason people can't pool money with a private agreement, with corporate charters being only between the corporation and its investors). There are examples of corporation-like entities, complete with stock markets, that existed without government intervention out there. Also, see "Bitcoin" and "Colored Coin" and possibly OpenTransactions.
Commercial Transaction Laws - Who would sell goods if they couldn’t be sure they would be paid, and who would buy goods if they couldn’t be sure they would receive them?
Everyone who is currently buying and selling on eBay? The Uniform Commercial Code doesn't need a government or legislature to be created or amended, just an agreement between business owners. Nor does it need a government enforce it, any more than an eBay member with a bad rating needs eBay to intervene to stop the member from selling.