Can bodyweight exercise alone build muscle?

Short answer

Yes. Push-ups alone thickened the chest about 18%, close to bench pressing. Legs grew less than with barbells (3.6% vs 6.3%), though, and heavy weights did better for raising your max lift.

Can bodyweight exercise alone build muscle?
'No weights → weak stimulus → no growth' — does this formula hold?
When you cannot get to a gym, it is natural to wonder whether push-ups and squats alone can build muscle. We checked step by step using studies that compared them directly with barbells.
Push-ups vs bench press
The chest thickened by about the same
18 people trained 2 times a week for 8 weeks: one group did a light bench press, the other did push-ups adjusted, for example on the knees, to the same load. Chest thickness rose 18.3% with push-ups and 19.4% with bench press, and both groups raised their max bench.
Does it work for experienced lifters?
Make it hard, and strength rises too
30 experienced lifters spent 5 weeks doing either a bench press they could lift only 6 times, or push-ups with a rubber band set so they could do only 6. Both raised their max bench by a similar amount. In another study, 23 people doing harder and harder push-ups for 4 weeks also raised their max bench.
What about suspended push-ups?
Back of the arm 16% thicker, abs 27%
8 people did push-ups on suspension straps 3 times a week for 8 weeks, 3 sets each until they could not do another. The back-of-arm muscles got 16% thicker and the abs 27%. Their max bench, however, did not rise.
How can muscle grow without weights?
Near your limit, light loads still build muscle
An analysis that pooled 21 studies found that when every set went until no more reps were possible, light and heavy loads grew muscle by the same amount. The heaviest weight you can lift once rose more with heavy loads, though. Bodyweight training works on the same principle.
But legs are harder
Thighs grew a little over half as much as with barbells
37 people trained 2 times a week for 8 weeks. The bodyweight group's thigh muscles grew too, but by 3.6%, less than the 6.3% in the barbell group. They grew less even though the bodyweight group started with smaller thighs, and only the barbell group gained meaningful thigh strength.
Bottom line: bodyweight alone builds muscle
The rule: near your limit, and keep making it harder
Push-ups built the chest about as well as the bench press. Going until you cannot do another rep, and making it harder by changing position or adding a band, works. For big leg growth or a higher max lift, heavy weights did better.