(Updates prices, adds reaction to U.S. Senate postponing health vote)
* U.S. yields rise with European debt after ECB's Draghi comments
* Trump agenda in focus as U.S. Senate delays healthcare vote
* Yellen says some assets are rich, Fed has eye on inflation
NEW YORK, June 27 (Reuters) - The euro rallied against the U.S. dollar on Tuesday after European Central Bank President Mario Draghi fueled market expectations the ECB will reduce stimulus later this year, while the dollar's weakness helped lift crude prices.
The dollar weakened further and stocks fell after a planned U.S. Senate vote on a healthcare revamp bill was postponed, while Federal Reserve Chair Janet Yellen said asset valuations are somewhat rich.
The delay in the healthcare vote sent U.S. stocks to session lows as it brought back worries about the time table of President Donald Trump's business-friendly agenda. More time spent on healthcare pushes back the discussion on a tax reform eagerly eyed by investors.
The euro hit a 10-month high versus the greenback at $1.1349 in afternoon trading in New York, but most of the move came earlier after Draghi, speaking to a conference in Portugal, said the ECB could adjust its policy tools as economic prospects improve in Europe.
"Just the fact that the ECB is considering their options right now is considered to be a hawkish signal," said Sireen Harajli, FX strategist at Mizuho in New York.
The dollar index fell 1.09 percent, with the euro up 1.5 percent to $1.1347.
But the Japanese yen weakened 0.26 percent versus the greenback at 112.14 per dollar.
Sterling was last trading at $1.2824, up 0.82 percent on the day.
The dollar weakness helped boost crude futures prices, though the backdrop of a long-standing supply glut kept gains in check.
U.S. crude rose 2.07 percent to $44.28 per barrel and Brent was last at $46.73, up 1.96 percent on the day.
Tim Evans, Citi Futures' energy futures specialist, said in a note that oil's move was "a technical correction after the declines of the past five weeks" helped along by boosts from a weaker dollar and forecasts for a weekly draw in U.S. crude inventories.
On Wall Street, bank stocks helped limit losses in technology shares, which extended after the healthcare vote delay.
The market likes certainty and now theres uncertainty," said Peter Costa, president at trading firm Empire Executions Inc. "What is this (health bill) going to look like when this gets out of the next iteration? That uncertainty I think is just having people pause a little bit.
"I also think that when the market gets to certain levels, any type of uncertainty, especially in anything that has to do with the (Trump) administration, will have an adverse effect."
The Dow Jones Industrial Average fell 40.35 points, or 0.19 percent, to 21,369.2, the S&P 500 lost 12.1 points, or 0.50 percent, to 2,426.97 and the Nasdaq Composite dropped 78.15 points, or 1.25 percent, to 6,169.00.
The pan-European FTSEurofirst 300 index lost 0.69 percent and MSCI's gauge of stocks across the globe shed 0.22 percent.
Emerging market stocks lost 0.38 percent. MSCI's broadest index of Asia-Pacific shares outside Japan closed 0.21 percent lower, while Japan's Nikkei rose 0.36 percent.
U.S. Treasury yields rose in sympathy with European government debt weakness, after Draghi's comments.
He surprised the market with that upbeat stance, said Tom di Galoma, a managing director at Seaport Global in New York. The European government bond market didnt take it very well.
Benchmark 10-year notes last fell 18/32 in price to yield 2.1998 percent, from 2.137 percent late on Monday.
The Treasury yield curve continued to flatten, with the spread between five-year notes and 30-year bonds dropping to a low of 92.5 basis points, the flattest level since late 2007. The spread between 2- and 10-year notes widened slightly to 82.5 basis points.
Fed Chair Janet Yellen said it is appropriate to gradually raise rates and noted that the U.S. central bank is carefully watching inflation expectations.
In corporate news, the EU slapped a record 2.42 billion euro fine on Alphabet's Google saying it had abused its dominant market position. Google said it was considering an appeal. Alphabet shares fell 2.4 percent.
Gold prices, which tumbled to their lowest level in nearly six weeks on Monday, were supported by the softer dollar.
Spot gold added 0.5 percent to $1,250.04 an ounce. U.S. gold futures gained 0.32 percent to $1,250.40 an ounce.
Copper rose 0.90 percent to $5,846.50 a tonne.
(Reporting by Rodrigo Campos, additional reporting by Sam Forgione, Devika Krishna Kumar, Lewis krauskopf and Karen Brettell; Editing by Nick Zieminski)