* U.S. ISM manufacturing index shows contraction in June * Euro zone factory PMI holds at 2-1/2-year low * PMIs fall across Asia to multi-month lows, except India * Exports orders falling in U.S., China, Korea, Taiwan By Yati Himatsingka and Jason Lange LONDON/WASHINGTON, July 2 (Reuters) - Europe's debt crisis slammed into the world's factories last month, with U.S. manufacturing contracting for the first time in nearly three years and Asian countries hit hard by crumbling orders from abroad. Some of the most grim data released on Monday was from the euro zone. The jobless rate in the currency bloc rose to a record high in May while a measure of factory activity held steady at its lowest level since June 2009. Other factory surveys - from the United States to China and Brazil - suggested Europe's woes were reverberating throughout the global economy. "There is no doubt that there are common driving factors now in the global slowdown and the euro area is probably the most dominant one," said Jeavon Lolay, global economist at Lloyds Banking Group. "It is hitting confidence, it is hitting exports and it is probably hitting credit as well and bank lending." The U.S. factory sector was dragged down in June by a plunge i n new orders and a sharp drop in exports, the private Institute for Supply Management said. The ISM's index of factory activity fell to 49.7, just below the 50 mark that signals growth. "This is the biggest sign yet that the U.S. is catching the slowdown that is well underway in Europe and China," said Paul Dales, an economist with Capital Economics in London. The weak U.S. factory data puts pressure on President Barack Obama ahead of his November re-election bid, and could fuel expectations of the Federal Reserve easing monetary policy further at its next meeting, which ends on August 1. In Europe, Markit's Eurozone Manufacturing Purchasing Managers' Index (PMI) was unchanged at 45.1 in June, its lowest reading since June 2009. Manufacturing activity in Germany and Spain contracted at the fastest pace in almost three years, and while French and Italian PMIs rose slightly, they were still below the 50 mark. Released after a European Union summit where leaders agreed to help Spain and Italy borrow money, the euro zone data highlighted the problems policymakers face to restore the region's economic fortunes. Around 17.56 million people were out of work in the 17-nation euro zone in May, or 11.1 percent of the working population, a new high since euro area records began in 1995, the EU's statistics office Eurostat said. "Unemployment will continue to rise until we see an improvement in the economy, and that may not be until next year," said Steen Jakobsen, chief economist at Saxobank. The jobs component of the euro zone PMI showed manufacturers cut staff at the fastest rate in two and a half years in June. This and the PMIs added weight to expectations that the European Central Bank will c ut interest rates on Thursday to a re cord low of 0.75 percent. EXPORTING TROUBLE Britain's manufacturing sector contracted for the second straight month, albeit at a slower pace, reinforcing expectations the Bank of England will pump more cash into the struggling economy. Europe's economic ailments, triggered by a festering debt crisis, left Asian factories reeling in June. The HSBC Chinese factory PMI showed factory activity shrank at its fastest pace in seven months in June. The index slipped to 48.2 from May's 48.4. "The further decline in the output, new orders, new export orders components suggests that the China economy still faces downside risks in the near term," said Haibin Zhu, a JPMorgan economist in Hong Kong. He said he expected Beijing to further ease monetary policy in the months ahead. The run of weak data suggested no immediate pick-up for the world's second-biggest economy, a story that is similar in Japan, home to big-brand exporters, such as camera and printer maker Canon Inc, which earns about 80 percent of its revenues abroad. Japan's June PMI, released on Friday, slipped to 49.9. Its index for new export orders dropped to 47.5, the sharpest pace of contraction since February. The PMIs for South Korea and Taiwan showed their factory sectors contracting for the first time in five months. Latin America is also feeling the pinch. Manufacturing in Brazil, the largest economy in the region, shrank for the third straight month in June. Some countries are faring better, at least for now. In Mexico, demand from the United States led manufacturing to step up its pace of expansion last month. And in India, where the economy is more reliant on domestic activity, the factory sector picked up in June. Still, growth in new export orders in India was the weakest in seven months.