Will the cross-country air fare war -- currently confined to routes between New York and the West -- be extended to Washington?

Yesterday, Western Airlines fired what could be the first shot in such a battle if other airlines follow.Western announced that it wll offer travelers to Los Angeles a special $299 roundtrip fare starting next Sunday. Travelers have to leave for California before July 15 and make the return trip anytime before Aug. 15.

Western offers two daily round trip flights between Washington National Airport and Los Angeles with one stop in Minneapolis/St. Paul. Travelers may find the one stop well worth the savings. Western's $299 fare is the least expensive through-plane service to California next to World Airways' $280 round-trip fare. The regular coach fare to California is $568 round trip on the major carriers; Super Saver fares, with a seven-day advance purchase and other restrictions, range from $346 round trip at night to $377 round trip during the day.

There are no advance-purchase or minimum- or maximum-stay requirements on the Western fare, but it is

For passengers originating in Washington, Western's special fare also offers a free stopover in Las Vegas, Phoneix, Salt Lake City, San Diego or San Francisco.

No other airline yesterday leaped to match Western's initiative. Western's one-stop service may not be considered enough of a competitive threat to provoke a response. The airlines did not drop their prices in response to World's Washington-California service either.

The precipitating factor in the New York price war was the entry last week of Eastern Airlines into the transcontinental market with significant numbers of additional seats -- seven roundtrip flights a day aboard wide-bodied L-1011s -- at prices significantly below its major competitors.Eastern started with roundtrip fares ranging from $268 to $390, but dropped its fares to $99 each way without restriction. All the major carriers are now offering the same fare -- at least through the month of June.