At the most basic level, Argentina needs the bond market more than the bond market needs Argentina. Argentina has, in the past, tried printing its way out of debt - but at the cost of ruinous inflation. By continuing to litigate, Argentina is seeking to reassure investors that their most recent default was not Argentina's fault, and that further lending should not be curtailed.
The background, from The Economist is summarized well here:
The countdown started on June 16th, when the Supreme Court of the United States announced that it would not get involved in Argentina’s battle with NML Capital, a hedge fund that picked up cheap debt after Argentina’s 2001 default and has since litigated for payment of full principal plus interest. The decision left intact a ruling by Thomas Griesa, a New York district-court judge, which banned Argentina from paying the creditors who in 2005 and 2010 swapped 93% of defaulted debt for performing securities, if the country did not also pay NML what it wants.
Argentina was left with only thorny choices: pay NML the $1.3 billion plus interest awarded by Judge Griesa; negotiate a settlement with the hedge fund; or stop paying the exchange bondholders. A payment due on June 30th to those exchange bondholders was missed. The grace period expires on July 30th, at which point Argentina will again be in default
Full payment would be hard to swallow. President Cristina Fernández de Kirchner has always opposed stumping up, and a generous payment for NML would open the door to similar payouts to other “holdout” creditors. The consensus has been that Argentina would reach a negotiated settlement with the holdouts. Argentina had made good progress this year in its quest to regain access to capital markets—settling disputes with the Paris Club of government creditors, for example. Tackling the holdout issue was the logical next step. The price of dollar-denominated defaulted debt surged above face value after the Supreme Court’s ruling, opening up a gap with euro-denominated debt that is not subject to New York law (see chart).
There is much more on the case from The Economist here, but the following quote is probably the most helpful:
But the costs to the country will rise the longer it remains in default. “The situation might not be cataclysmic, but defaulting was still the worst decision the government could have made,” says Maximiliano Castillo of ACM Consultants. The country may have got used to doing without access to external financing, but its foreign-exchange reserves have been shrinking, and the economic boost it received in the 2000s from rising commodity prices will not be repeated. It needs to borrow to grow.
The government is aware of this. In the past few months it has scrambled to resolve disputes with the International Centre for the Settlement of Investment Disputes (ICSID); the Paris Club, an informal group of government creditors; and Repsol, a Spanish oil firm whose stake in YPF, the state oil firm, it expropriated in 2012. A default will undo this progress.
The fear now is that the need to print money to finance its deficits will further spur inflation, putting the exchange rate under renewed pressure. Add in the likely jolt to consumption from default, and the recession that the country entered earlier this year will only deepen. Abeceb.com, a consultancy, is predicting that Argentina will contract by 3.5% in 2014; if the country had avoided default the projection was of a milder tightening of 1.5%.
Almost every major financial institution has a clause that says their disputes will be meditated and judged by the laws of the United States (e.g. New York) or the United Kingdom (e.g. London). While Argentina could try going to London, the actors are aware of each other, and nobody will want to lend to a country that doesn't abide by international norms.