Race stewards didn't overturn Ferrari's 1-2 finish, choosing to send the case to the sport's governing body, which could impose more sanctions. Team orders that affect the result of a race are forbidden under Formula One rules.
"In the interests of the sport, we have decided not to go through a procedure of appealing against it, confident that the [FIA] world council will know how to evaluate the overall facts correctly," Ferrari team chief Stefano Domenicali said.
Alonso had more points in the title race and the team apparently felt it would be better served if he collected the 25 points that go to the winner rather than Massa.
"I don't think I have to say anything to that," Massa said when asked after the race about the instructions. "We work for the team."
Massa, racing on the first anniversary of a crash that nearly killed him, led the race for 49 of 67 laps before he was passed by Alonso following communications from Ferrari over the team radio. He looked unhappy after the race and the two drivers barely hugged as they climbed out of their red cars.
Stewards said Ferrari was in breach of Article 39.1 of the International Auto Federation (FIA) 2010 sporting regulations.
Article 39.1 says: "Team orders that interfere with race results are prohibited."
After 11 of 19 races, Hamilton leads the overall standings with 157 points ahead of McLaren teammate Jenson Button with 143. Vettel has 136 points, tied for third with Red Bull teammate Mark Webber. Alonso is fifth with 123 points, 38 ahead of Massa.
"I can't say I am fighting for the championship," Massa said.
Alonso won his second race of the season and Ferrari finished 1-2 for the second time this year.
"In some parts of the race we were fighting very hard for first place, maybe it was a bit dangerous. It's a difficult race to overtake," said Alonso, a two-time Formula One champion who earned his 23rd career victory. "We are professional, we try to do the best for the team.
"I don't know what happened, but at the exit of turn six I saw Felipe a little bit slow. Sometimes you are quick, sometimes you are slow, and in some parts I was quicker than him, so it's very difficult to judge."
Red Bull team principal Christian Horner said Ferrari appeared to impose team orders.
"It was surprising what happened with the Ferraris, it looked to be a team order with the cars switching positions," Horner said. "If so, it's a shame for Formula One and the fans that they were deprived of a race between the two Ferrari drivers today."
The first of two decisive moments in the race took place at the start.
Pole sitter Vettel moved wide to try to block Alonso. Massa, starting from third, used the gap to pass from the outside going into the first corner, where also Alonso managed to slip past Vettel. The two Ferraris stayed in front for the entire race.
Then, on lap 49, Massa appeared to go slightly wide at a corner and Alonso easily shot past the Brazilian driver.
Alonso had been pushing Massa hard for several laps and was heard saying on the team radio, "This is ridiculous."
Alonso then overtook Massa, who had just been told by Ferrari over the radio: "Fernando is faster than you, did you understand that message?"
Having dropped into second, Massa was then told: "OK, good lad. Just stick with him now."
Massa confirmed the radio conversation, but later also said he had been struggling after switching to harder tires.
Massa said Ferrari did not have team orders.
"For sure you always want to win. We don't have team orders. If you can't do the race you want, you have to think about the team. I am professional, and today I showed how professional I am," Massa said.
"Everyone saw that I can win races, that I can be competitive. But I was struggling on hard tires."
In later comments distributed by Ferrari, Massa said it was his decision to let Alonso pass.
"We drivers have to first of all think of the interests of the team and that is what I showed again today. In my opinion this was not a case of team orders: my engineer kept me constantly informed on what was going on behind me, especially when I was struggling a bit on the hard tires.
"So I decided to do the best thing for the team, and a one-two finish is the best possible result."
Alonso covered the 67 laps -- a distance of 190.5 miles -- in 1 hour, 27 minutes, 38.864 seconds. He finished 4.1 seconds ahead of Massa and 5.1 seconds ahead of Vettel.
Asked about the passing maneuver of Ferrari, Vettel said: "We get the check not from you guys but from the team."
Alonso, in his first season with Ferrari, won the opening race of the season in Bahrain, where Massa took second.
But Ferrari has struggled in recent races and Alonso finished 14th in the British GP two weeks ago.
Massa required surgery for a fractured skull sustained one year ago when he was struck on the helmet by a heavy metal spring that had come off another car during qualifying for the Hungarian Grand Prix.
He was placed in an induced coma and spent nine days in a Budapest hospital.
Massa returns to Hungary for next week's Grand Prix.