While he's certainly Marvel's most popular character, Spider-Man has never been among the most powerful heroes. Peter Parker tends to handle smaller threats, including street-level crime and basic supervillains.
Despite his modest power-set, Spider-Man once took down two Phoenix hosts at the same time.
This stunning achievement took place in Avengers vs. X-Men #9, by Jason Aaron, Brian Michael Bendis, Ed Brubaker, Matt Fraction, Jonathan Hickman and Adam Kubert. At this point in "Avengers vs. X-Men," The Phoenix Force was split between five hosts, known as the Phoenix Five.
The Phoenix's power naturally corrupted these X-Men, however, leading Namor to destroy Wakanda just to capture the Avengers. It took all the power at their disposal for Earth's Mightiest Heroes to defeat Namor. Afterward, his power transferred to the other four hosts. The Avengers fled to K'un-Lun, where Spider-Man trained Hope Summers so she would eventually claim The Phoenix for herself. During this time, many Avengers were defeated and captured by the X-Men and the Phoenix hosts.
Fortunately, Storm decided to help the Avengers, leading them to the Verkhoyansk Mountains to free their captive teammates. Arriving inside the fiery mountain, the Avengers were instantly detected by Colossus and Magik. Earth's Mightiest Heroes were cornered, taking on two Phoenix hosts at once.
Spider-Man quickly realized that there was no way the Avengers could win this fight. Allowing his teammates to escape, Peter decided to stall the Phoenix siblings. Of course, the web-slinger didn't stand a chance against these two physically. Yet Spider-Man kept at it, fighting the Phoenix hosts long after Colossus told him to stay down.
The fight was brutal, leaving Spider-Man with several broken bones, a bloodied face, crushed web-shooters, and a shredded costume. Peter even noted that he was beginning to lose feeling in his head. While Colossus held back, Magik became frustrated with her brother.
Illyana questioned why Piotr didn't simply finish the wall-crawler. In turn, Colossus responded that he was worried that The Phoenix was corrupting his sister. Spider-Man used this discord to his advantage. Peter reminded the mutant siblings that, if one of them fell, the other would absorb their power. After a brief glance, the two mutants attacked each other. By the time the rest of the Avengers arrived to rescue Spider-Man, they found Colossus and Magik unconscious, each one having dispatched the other.
Spider-Man's victory is likely to have made him an enemy of The Phoenix. Considering Peter's absence from the recent tournament to determine its next host, it's possible that The Phoenix didn't want the web-head there. For one thing, it must have been very painful for The Phoenix to turn against itself. Indeed, when Colossus and Magik fought, they each possessed a fragment of The Phoenix Force, making the cosmic entity harm itself during their eruptive skirmish. This battle must have been very confusing and painful for The Phoenix, an ordeal for which it probably blames Spider-Man.
The Phoenix also underestimated Spider-Man, discovering how unpredictable the wall-crawler could be. Peter's wit and guile, which he used to take down Colossus and Magik, are probably not the traits that The Phoenix desires in a new host. If Peter was The Phoenix's next host, there's no telling what trick he might try to pull.
Additionally, Spider-Man's ironclad will, which he also demonstrated in the fight, would also clash with The Phoenix. If Peter became the next Phoenix host, he'd probably fight against this cosmic entity, leading to undesirable outcomes. While others, like Captain America, have this same willpower, The Sentinel of Liberty didn't harm The Phoenix the way Spider-Man did.
Ultimately, after defeating two Phoenix hosts, it's safe to say that Spider-Man is not one of The Phoenix Force's most desired humans.