Berger also believes the double world champion, 33, will spend the rest of his career at the Surrey-based team.
Alonso has been ruled out of the first race of the season in Australia on 15 March following his crash in testing.
Asked if he expects Alonso to win the title at McLaren, Berger, who drove for the team from 1990-92, told BBC Sport: "Yes - he is a complete driver."
He added: "He should have more championships because he is a big champion. The way that he drives, the killing instinct that he has, he's capable of winning more than two championships."
Spaniard Alonso said in a video message on Friday that he is "completely fine" following the accident at Barcelona on 22 February that left him with concussion.
However, doctors advised him not to race in Melbourne because of the risk of suffering a second concussion.
Berger, the winner of 10 races in an F1 career that lasted from 1984-97, says Alonso, world champion in 2005 and 2006, will have no physical or mental problems once he returns to the track.
"Crashes are part of the game in F1," he said.
"Fernando is such an experienced driver, he will deal with it in a good way and I don't think it's going to have any effect on him. He will be ready to go racing again."
McLaren completed the fewest miles of the nine teams who took part in pre-season testing, as they and new engine partner Honda struggled with reliability issues.
Berger, who drove for McLaren when they previously used Honda engines, expects 2015 to be "a learning year" for the team but believes the Japanese company is "going to get the job done".
He predicts that McLaren will be challenging for race wins in 2016 and targeting the drivers' championship after that.
Berger added: "In my time together with McLaren and Honda, it was extremely professional, extremely competent. There was a big hunger for success so I think this is the culture of Honda.
"I don't think it should matter if tests at the moment are not great or if the first races are not great. It's normal - it would be more of a surprise if Honda were competitive from day one."
The 55-year-old, who has had senior managerial roles in F1 with BMW and Toro Rosso since retiring as a driver, says Alonso should have between three and five years left of his own racing career and believes he will win at least one more title before he finishes.
Berger also expects Alonso, who has joined Britain's 2009 world champion Jenson Button at McLaren, to stay with the team for the rest of his F1 career.
"I think once he signed up for McLaren-Honda, he knows there is going to be a learning period. I'm sure that he is going to be part of this learning," he added.
"Once this is done, I'm sure he wants to take all the fruits from the learning period. I don't think it must be an intention to jump around teams.
"Alonso has some extreme high talent in certain areas, not just in driving skills but in the way he runs the politics."
Fernando Alonso will win the world drivers' title at McLaren, says their former driver Gerhard Berger.