* U.S. stocks end lower

* Facebook falls as it faces U.S. lawsuits

* European stocks end up 0.3%

NEW YORK, Dec 9 (Reuters) - Global equity indexes fell on Wednesday as negotiations over further U.S. fiscal stimulus dragged on and the dollar rose for a fourth straight session.

All three major U.S. stock indexes ended lower, with the Nasdaq down 1.9% and leading the day's losses.

Shares of Facebook, among the biggest drags on the S&P 500 and Nasdaq, fell 1.9% after the U.S. Federal Trade Commission and nearly every U.S. state sued the social media company on Wednesday, saying it broke antitrust law and should potentially be broken up.

Investors were awaiting news on further U.S. economic relief as the coronavirus pandemic continued to take its toll on the economy.

U.S. Senate Majority Leader Mitch McConnell said lawmakers were still looking for a path toward an agreement on COVID-19 aid, as the House of Representatives prepared to vote on a one-week funding bill to provide more time for a deal.

"We had a pretty nice boost in the morning on speculation that we might get a new package, but it's kind of an on-again and off-again thing," said Randy Frederick, vice president of trading and derivatives for Charles Schwab in Austin.

The Dow Jones Industrial Average fell 105.07 points, or 0.35%, to 30,068.81, the S&P 500 lost 29.43 points, or 0.79%, to 3,672.82 and the Nasdaq Composite dropped 243.82 points, or 1.94%, to 12,338.95.

The S&P 500 had hit a record high early in the session.

The pan-European STOXX 600 index ended up 0.3%. MSCI's gauge of stocks across the globe shed 0.46%, also hitting a record high earlier.

Positive news on COVID-19 vaccine development has been boosting stocks. Johnson & Johnson this week said it could obtain late-stage trial results for a single-dose vaccine in January, earlier than expected.

Britain this week became the first Western nation to begin a wide vaccination campaign. However, Britain injected a note of caution, saying people with a history of significant allergic reactions responded adversely to the Pfizer vaccine.

The dollar rose as stocks came under pressure. In afternoon trading, the dollar gained 0.1% against a basket of currencies to 91.027. It reached an April 2018 low of 90.47 last Friday.

British Prime Minister Boris Johnson began a crisis meeting over dinner in Brussels with the EU's chief executive on Wednesday, hours after warning the bloc it must make concessions to clinch a Brexit trade deal and avoid a turbulent year-end breakup.

Sterling was last trading at $1.3409, up 0.10% on the day.

U.S. Treasury yields rose. Benchmark 10-year notes last fell 8/32 in price to yield 0.94%, from 0.91% late on Tuesday.

Oil futures were little changed, although data showed an unexpected jump in U.S. crude stockpiles. Brent crude rose 2 cents to settle at $48.86 a barrel. U.S. crude fell 8 cents, or 0.2%, to settle at $45.52.

Spot gold prices were also near flat.

(Reporting by Caroline Valetkevitch; Additional reporting by Chuck Mikolajczak and Laura Sanicola in New York and Shriya Ramakrishnan in Bengaluru; editing by Chris Reese and Nick Zieminski)