(Recasts headline to reflect U.S. stocks' gains)
* Euro zone stocks fall further, Wall St holds small gains
* Treasury yields edge lower
* Markets seen volatile until Greek referendum
By Rodrigo Campos
NEW YORK, June 30 (Reuters) - Stocks on Wall Street rose in choppy trade on Tuesday and a global equities gauge ticked up, while the euro remained under pressure as Greece grappled with a looming deadline on a debt payment to the IMF.
Greece submitted a new two-year aid proposal to its creditors, calling for debt restructuring in what seemed like a last-ditch effort by Athens to resolve its impasse with lenders. Talks between Greece and its creditors broke down over the weekend, causing Greece to close its banks and impose capital controls.
Greeks are due to vote in a referendum on Sunday that EU partners say will amount to a choice between staying in the euro or leaving.
German Chancellor Angela Merkel played down any hopes of a last-minute deal with Greece on Tuesday and said there will be no new negotiations before the planned referendum.
"Probably the market still has in the back of its mind the idea this is all a bit of a game and eventually they'll come with some sort of accommodation to avoid a worst-case scenario," said Darren Williams, AllianceBernstein's London-based senior economist for Western Europe.
"There is a certain complacency," he said, adding that he expects opinion polls on the Greek referendum vote to keep markets volatile.
The CBOE Volatility Index rose for a fifth straight session, its longest streak since mid January.
The Dow Jones industrial average rose 74.27 points, or 0.42 percent, at 17,670.62. The Standard & Poor's 500 Index was up 10.79 points, or 0.52 percent, at 2,068.43. The Nasdaq Composite Index was up 38.28 points, or 0.77 percent, at 4,996.75.
The S&P earlier rose as much as 0.8 percent.
The MSCI All-Country World equity index was flat. The pan-European FTSEurofirst 300 index and the euro zone's blue-chip Euro STOXX 50 index were down 1.3 percent each.
The borrowing cost on a key source of overnight loans for Wall Street jumped as traders competed for a shrinking pool of cash before quarter end. The interest rate on overnight loans in the repurchase agreement market was last quoted as high as 0.65 percent, which would be the highest closing level since November 2008.
In commodities, oil futures bounced back from three-week lows but Brent was set to close its second consecutive month of losses.
"Markets are worried that a Greek debt default could hit European economic growth and thus fuel demand," said Tamas Varga, oil analyst at London brokerage PVM Oil Associates.
Brent was up 2.2 percent at $63.40 a barrel after falling to $61.35 on Monday, its weakest since June 5. It was down 3.7 percent for June. U.S. crude gained 1.4 percent at $59.15. It was set for its first monthly decline in three, down 2.2 percent.
The euro was last down 0.8 percent against the U.S. dollar at $1.1149.
The euro's weakness is "partially the reality that there is not going to be a (Greek) payment to the IMF," said Jason Leinwand, managing director at rates, currencies and commodities derivatives hedge advisory firm Riverside Risk Advisors in New York.
He said, however, that optimism for a Greek resolution lingered, given the upcoming referendum.
Treasuries yields edged lower. Benchmark 10-year Treasury notes were off 1/32 in price to yield 2.333 percent.
The 30-year bond was unchanged in price and yielding 3.097 percent. (Reporting by Rodrigo Campos, Richard Leong, Barani Krishnan and Sam Forgione in New York, additional reporting by Lionel Laurent and Christopher Johnson in London; Editing by Dan Grebler)