No, Lanza never sang in any of the major opera houses, though he certainly could have if he had wanted to. Throughout his life, he received well-documented offers from the Met, La Scala, and the Rome Opera, just to give a few examples. The directors of these theaters all heard Lanza, and were satisfied that he had both the voice and the musicality to perform at the highest level. To quote Marilyn Horne, who sang in the Roger Wagner Chorale when Lanza appeared at the Hollywood Bowl in 1948: "That sound! Man, he was the real thing. He would have been a spinto tenor. He *was* a spinto tenor."
Or Richard Bonynge and Joan Sutherland, who heard Lanza perform (unamplified) at the Royal Albert Hall: "We were both surprised by the size of the voice. . . . We were also impressed by Lanza's innate musicality. No doubt he could have had an outstanding operatic career."
Why didn't Lanza enjoy a great operatic career? Essentially because he lost his nerve. He'd started out in the best possible way, earning rave reviews from the likes of Opera News and the New York Times for his operatic debut at the prestigious Tanglewood Music Festival in 1942. At just 21, he was acclaimed as having "few equals among tenors of the day in quality, warmth, and power." That was the New York Times. Army service interrupted his career, but on his return to civilian life, he studied hard with the best of teachers (Enrico Rosati), resumed his concert career (gaining excellent reviews from some of the toughest US critics), and did a couple of extremely well-received Pinkertons for the New Orleans Opera Association.
But by then he had effectively sealed his fate by signing a 7-year contract with MGM. From then on, there was never enough time for him to juggle a film career with an operatic one, despite his insisting on a contract that gave him six months off a year for concerts, opera appearances, etc. Hollywood was the worst possible environment for someone so vulnerable, and a person who always needed discipline in his life. The pressures of stardom---and *especially* the growing skepticism from music & armchair critics who had never heard him in live performance and poured scorn on his talent because he was a film star---got to him. The pressure & criticism he was exposed to was unprecedented for a young operatic singer. Ridiculous myths about the size of his voice and even his ability to learn operatic roles began circulating, and still endure to this day. In short, the super-sensitive Lanza lost faith in himself, and afraid of failing on the stage, where he would have been exposed to incredible scrutiny, he kept delaying his return to opera. And he began drinking heavily (though he never had a drug problem, as has been suggested here).
Near the end of his life, however---after he finally left Hollywood and moved to Italy---there were signs that Lanza could have pulled himself together and returned to opera, "his one true love," as Peter Herman Adler wrote. He started performing in recital again (throughout Europe), received excellent reviews, and in the last year of his life was working for 2 hours a day with a vocal coach. He even agreed to sing the role of Canio in Pagliacci at the Rome Opera (having recorded some substantial operatic selections there the previous year, and impressing artistic director Riccardo Vitale). But then he died.
So how do we assess Lanza the artist? It's not valid to compare his limited operatic career with the careers of Tucker, Di Stefano et al (though it's worth noting that both those singers held him in very high regard). It *is* valid, however, to compare him with other singers on the basis of his recordings---some of which hold up very well indeed against the biggest names (if one is prepared to hunt for them), and a few of which arguably even surpass the efforts of others. (Listen to Lanza's recordings of the Improvviso from Andrea Chenier, M'Appari' from Martha, and the Otello Monologue for starters.)
And if you're *still* skeptical about Lanza, and his place among the great voices of the last century, read the opinions of people far better qualified than anyone here to judge an outstanding operatic talent: great singers like Robert Merrill, Licia Albanese, and Dorothy Kirsten, all of whom either worked with him or heard him in live performance: http://www.mariolanzatenor.com/quotes-from-opera-singers.html
ETA: In response to Richard: Lanza's problem wasn't stamina. His technique was such that he could sing for hours without tiring, as the Met soprano Dorothy Kirsten observed, and in the 1940s he proved he could handle the most grueling of performing schedules, singing 86 concerts of demanding operatic fare in a 10-month period. See the press reviews available on my website: http://www.mariolanzatenor.com/concertsopera-1942-1950.html.