Tampons are super absorbent and tend to expand quite a bit in the juices of the gastrointestinal system. It may pass but it is also possible that it could cause an obstruction.
This long after ingestion it is too late to induce vomiting, as it is reasonable to think most of the pieces have made their way into his intestinal tract.
I wouldn't panic quite yet, as he seems to feel fine and has only vomited once. That may just be the irritant effects from eating foreign material.
To try and settle his stomach tonight you can give either:
1) Pepcid-ac (famotidine) at a dose of one half of a 10mg tablet per 10 to 20 pounds of body weight every 12 hours.
2) Prilosec (omeprazole) at a dose of one half of a 20mg tablet per 20 to 40 pounds of body weight every 24 hours.
These will reduce stomach acid and should help settle his stomach. They can be given for several days if need be.
Then feed a homemade diet of 1/3 boiled, lean hamburger or boiled, white, skinless chicken, all fats and juices drained off the meat, and 2/3 boiled, plain, white rice. Add 2 tablespoons of canned pumpkin, (not pie filling, just pumpkin), to each meal for fiber to help push the remaining pieces through.
All you can do now is watch him closely. If he starts vomiting several times and cannot keep food down, has a tense painful belly or refuses to eat or runs a fever (more than 103F rectally) then that can signify a blockage and he should be seen by a veterinarian immediately. Surgery may be needed at that point.
Please let me know if you have any further questions.