The voters of Texas sent Gov. Mark White a message Saturday, forcing the incumbent to sweat out a narrow Democratic primary win against a field of obscure challengers. The question now is whether that protest vote, prompted in large part by unhappy teachers and public employes, will carry over to the general election campaign.

That campaign will pit White against former governor William Clements, who easily secured the Republican nomination.

From the start this year, White and his strategists said they would rather have a rematch with Clements, the 69-year-old Dallas oilman whom White defeated in 1982, than with either of the other two potential Republican candidates -- Rep. Tom Loeffler, a rising power in the House leadership, and Kent R. Hance, the former west Texas congressman who switched parties last year. White got his wish, as Clements received 59 percent of the Republican vote, with Loeffler and Hance winning about 21 percent apiece.

Against Clements, who leads White in early polls by as much as 12 points, White can play the underdog, a campaign posture he finds comfortable.

"He relishes being down, being the fighter," consultant George Shipley said. "He can set up the campaign almost as though old Bill Clements is the incumbent and Mark White is the scrappy challenger."

To accomplish that, White must link Clements to the Reagan administration, and then persuade voters that the Republicans are responsible for the state's economic troubles. Slumping oil prices have raised the unemployment rate in Texas to 8.5 percent and created a $1.3 billion shortfall in the state budget.

"White has to make the case that Republican inaction is responsible for the state's troubles," said Richard Murray, a pollster and political scientist at the University of Houston. "He's tried to do that in recent months with mixed results. One-on-one with Clements, it might be easier."

According to Murray's analysis, White's troubles in the primary -- he received only 53.6 percent of the vote and ran 20 percent to 30 percent behind the rest of his ticket -- reflected a yearlong trend in which the governor's political base has deteriorated. Murray found that about half the Democrats who were disaffected with White voted for other candidates in the Democratic primary, and half switched to the Republican primary.

"There were a lot of defections, which shows White's vulnerability," Murray said. "Clements, on the other hand, had virtually no defections. It might be too early to call White the underdog, but there is a definite initial leaning in the state not to go with the governor."

It is less certain whether teachers and public employes who mounted an "anybody-but-White" movement during the primary will turn to Clements in November.

Clements is the only candidate who has vowed not to raise taxes under any circumstances, a promise that seems to put him at odds with the public employes' push for pay raises. And teacher dissatisfaction with mandatory competence tests may soon diminish. Test results are due Monday and, according to sources, they will show that 95 percent of the teachers passed, far more than predicted.

"White's been an activist governor, and when you do things you tend to upset people once in awhile," said Jim Hightower, state agriculture commissioner, who ran far ahead of White on the ticket of Democratic incumbents, receiving 81.5 percent of the vote. "Those groups sent him a message, but when they look at the race in November, a lot of them will come home. Sending a message is not worth four years of a Republican governor who doesn't believe that the state should have much of a role in trying to make things better for everyone," Hightower said.

One key factor in White's 1982 upset of Clements was the presence of Sen. Lloyd Bentsen on the ticket. There are no Senate races in Texas this year. Four years ago, Bentsen's money and organization were considered crucial to the success of the entire Democratic ticket. This year more of that burden will have to be assumed by Lt. Gov. William P. Hobby, who, like Bentsen, is a wealthy, conservative Democrat.

Clements, Hightower said, "has enough money to beat a wet mule," and likely will spend much of it in the general election campaign. Four years ago Clements spent $13 million, a Texas record.

The former governor's surprisingly easy victory over Hance and Loeffler was attributed to three factors.

First, he made no serious blunders in the dozens of debates and confrontations with his two challengers. In previous campaigns, Clements has appeared gruff and arrogant, but this time he was composed and relaxed.

He also benefited from a bandwagon effect after taking an early lead in the polls. Murray, the Houston pollster, found examples of Loeffler and Hance supporters who decided in the last week to vote for Clements so that the party would not have to endure a divisive runoff. "He probably picked up another 10 percent that way in the last week," Murray said.

Finally, the Republican voters found serious flaws in Hance and Loeffler. Hance based his campaign on a strong effort to attract conservative Democrats, but he never struck the right chord with mainstream Republicans. Loeffler was uniformly regarded as a nice guy but a disaster as a campaigner. One Republican said he needed a "charisma bypass."