I wouldn't consider myself the resident economist; Brandon has a better grasp for economics than I do. But, off the top of my head, here's what would probably happen if everyone decided to not pay their credit cards:
1. The credit card companies would not get their rightful income. Yes, credit card companies do collect a fee from the merchant for giving them the money right away, but that's only about 2-3% of the purchase price. Let's say you had a meal in a nice restaurant, and the total came to $100, which you charged. The credit card company would pay the restaurant $97, and then expect you to pay them $100 when your statement becomes due. If you did not pay them that $100, and had no plans to pay them that money, the card issuer is out the $97 they advanced to the merchant. That's not that big a deal if only a few people default, but if everyone defaults, that's a crisis.
2. The credit card companies don't hold all their assets themselves. They repackage credit card receivables into asset-backed securities. Essentially, they bundle together the credit card debt of thousands of people, then issue certificates on this debt to other investors. If the credit card company expects to get an average return of 7% on this debt, they might package it together and pay a 6.5% coupon. This gives the company a lower return, but also insulates it from the default risk by passing it on to other investors. If no one paid their credit cards, these investors are the ones who would get screwed. Given that credit card interest rates are significantly higher than Treasury rates, these bonds were likely sold at a premium, and when there is mass default, their prices will drop like a rock.
3. A mass default in the credit card sector will lead to a flight to quality. Basically, if everyone is defaulting on their credit cards, investors will see the environment as being much riskier than before, and will pull their money out of risky investments and into investments with less risk (generally AAA corporate bonds or Treasuries). This will cause spreads to widen on bonds with lower credit quality, and companies with riskier assets will try to unload them on the marketplace. Since everyone's trying to unload them at the same time, though, the market gets flooded, so the prices on these assets will drop. This would have a devastating effect on the financial health of companies that were holding riskier asset classes with low liquidity (like high-yield bonds or mortgage-backed securities), because they have to mark their assets to market, even if they are still holding these assets. If a company has $100 million in assets and $98 million in liabilities, it has $2 million in surplus, and is doing OK. If the value of its assets drops by 10%, now it has a surplus deficit of $8 million, and is in serious trouble.
4. Defaults on credit cards will cause investors to reconsider whether they had underpriced the risks of default for all products. There would be a mass pull-back of offering home loans. After all, if consumers are willing to default on their credit cards, why wouldn't they default on their mortgages? The screening and underwriting process for mortgage loans would tighten considerably, and there would be many fewer people who would qualify for home loans under the new guidelines. Chances are that you would need to put more than a 20% down payment on a house to get a loan (it would be more like 50%). Those who would qualify would see substantially higher mortgage rates due to the perceived increase in riskiness of the loans. Refinancing wouldn't be nearly as easy, and it's likely that people with high mortgage rates or ARMs would be stuck with them.
5. Given the potential for the deterioration of the massive financial services industry, there would probably be a congressionally-authorized bailout (think about the savings and loan industry fiasco, times a million). So, instead of the credit card consumers paying the credit card companies, the taxpayers would end up paying the credit card companies. Those who lived without credit cards get screwed because they have to pay for the irresponsible actions of others. Those who had small balances on credit cards get screwed as well, because the payment they will have to make to the government will exceed the outstanding balance they had on their cards. Those few who had giant balances will still have to pay something, but not nearly as much as they would have if this situation hadn't happened, so they're happy (provided that their neighbors don't beat them into a bloody pulp). You may have gotten away with not paying the credit card company, but you're not going to stiff Uncle Sam.