Hello, my name is***** and I have over 20 years of experience as a veterinarian. I'm sorry to hear that your pup ate something he should not have.
Tampons are super absorbent and tend to expand quite a bit in the juices of the gastrointestinal system. It may pass but since he just ate it, within the past 2-3 hours, it's worth trying to get him to vomit it back up.
You can give him 1 tablespoon of 3% hydrogen peroxide per 10 pounds of body weight mixed with bread chunks or ice cream (3 tablespoons for a fellow his size is fine).
Then take him outside and run him around to get the peroxide bubbling so he vomits.
If no vomiting in 10 minutes repeat the peroxide dose once.
If he vomits and seems to clear his stomach and be done I would check to see if the pieces are present.
Once he finishes vomiting and empties his stomach to try and settle his stomach today you can give either:
1) Pepcid-ac (famotidine) at a dose of one 10mg tablet per 20 to 40 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.
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 whether he vomits most of the foreign material back up or not to help push the remaining pieces through.
If you don't have peroxide you can get him to an emergency clinic to induce vomiting.
If he doesn't vomit it back up or you choose not to take him to an emergency clinic still use the bland diet with pumpkin to try and get the pieces to pass.
If he vomits but no pieces are seen then his stomach has emptied since he ate the tampon. Depending upon the dog the stomach usually takes 2-3 hours to empty, and may be slower with foreign, bulky material. So after 2 hours you may still have a chance and it is worth trying to induce vomiting. If the pieces are out of his stomach all you can do at that point is watch him closely. If over the next few days he starts vomiting, 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.