* Investors worry euro zone rescue only buys some time
* Doubts remain on how euro zone countries will cut deficits
* Euro dips, sheds gains made on EU's rescue package
* Asian shares fall 1 pct after best 1-day pct rise in a year
* Foreign investors sell Seoul, Tokyo shares
TOKYO, May 11 (Reuters) - A massive relief rally in financial markets on news of a $1 trillion deal to resolve Europe's debt crisis fizzled on Tuesday on nagging doubts about how Greece and other debt-laden euro zone countries will reduce their budget deficits.
Asian stocks fell more than 1 percent and European shares looked set to follow, with financial bookmakers expecting bourses in London, Frankfurt and Paris to open as much as 1.8 percent lower.
U.S. stock futures fell 0.9 percent, with early losses accelerating heading into the European market open.
The euro, which had initially surged on the news of the weekend package put together by EU finance ministers, central bankers and the IMF, eased on Tuesday and was well off its Monday high as investors grew cautious.
The single currency dipped 0.6 percent to $1.2705 by early afternoon from late U.S. levels on Monday.
The rescue package, together with a pledge from the European Central Bank to buy government bonds, helped ease worries that Greece's debt problems would engulf other weak countries in Europe and spark another global credit crisis.
In response, markets soared worldwide. Stocks in fiscally weak Greece, Spain and Portugal jumped by double digits and their risk premiums on their debt tumbled against benchmark German bunds. Wall Street racked up its biggest one-day gain in more than a year.
But while the safety net may have eased fears of a Greek debt default in the near term, longer-term concerns remained over whether Greece and other euro zone countries with large fiscal deficits will be able to deliver on tough fiscal austerity measures and avoid debt restructuring.
Moreover, the crisis has highlighted deep-rooted political divisions and structural economic problems across the 16-country euro zone and broader European Union.
"Even though one of the worst scenarios -- a Greek default -- has been avoided for now, in many ways solving the bigger problems have simply been postponed and new issues could emerge in places such as Portugal and Spain," said Nagayuki Yamagishi, a strategist at Mitsubishi UFJ Morgan Stanley Securities.
MSCI's index of Asia-Pacific shares outside Japan fell 1.2 percent after climbing 3.4 percent on Monday, its biggest single-day percentage gain since May 2009.
Japan's Nikkei average, which gained 1.6 percent on Monday, fell 1.1 percent on concerns about the longer-term euro zone outlook.
Hong Kong's Hang Seng index fell 1.4 percent after its biggest jump in five months on Monday, while South Korean shares slipped 0.4 percent.
Market players said profit-taking weighed on equities following Monday's sharp rally, with foreign selling picking up.
In Korea, foreign investors were sellers of a net 29.1 billion won ($25.71 million) worth of shares. Foreign orders placed before the open of Tokyo trade showed foreign investors had also been set to sell for a fourth straight day.
Investors in Asia also kept a close eye on economic reports from China for any clues on whether authorities will ramp-up policy tightening measures in coming weeks, potentially dampening corporate profits.
Data on Tuesday showed China's inflation rate rose to an 18-month high in April and bank lending topped expectations, but the full suite of monthly data showed an economy that was in robust health and not overheating as some have feared.
Mitsushige Akino, chief fund manager for Ichiyoshi Investment Management, said the uptrend in Asian shares from troughs hit in late 2008 following the Lehman Brothers collapse still seemed intact despite the sharp fall seen last week.
"But since last week's shock and turmoil is still clearly in people's minds, it will probably take a while for markets to recover," he said. Graphic on foreign flows in Korea: r.reuters.com/ryp53k
EURO ON BACKFOOT AGAIN?
In currency markets, the euro was well off its Monday high near $1.3100, and retreated back to where it stood late last week before the European Union's emergency package was unveiled.
The single currency had slumped to a 14-month trough of $1.2510 last week as fears of a euro zone debt crisis intensified and as some global credit markets showed increasing signs of strain.
"The emergency package is effective in avoiding a near-term crisis. But so much uncertainty remains for the euro zone," said Tsutomu Soma, senior manager of the foreign securities department at Okasan Securities.
Sterling fell 0.3 percent to $1.4811 held back by political uncertainty in Britain following an inconclusive election that left no single party with a clear majority to rule.
Prime Minister Gordon Brown said he would step down in an effort to try to keep his Labour Party in power. Labour and its major rival, the Conservatives, are trying to woo the smaller Liberal Democrats to form a government.
U.S. crude futures dipped 0.5 percent to $76.44 a barrel and spot gold edged higher to $1,205.10 an ounce. (Additional reporting by Elaine Lies, Rika Otsuka and Kaori Kaneko; Editing by Kim Coghill)