I also do not recommend inducing vomiting in cases where dogs eat sharp objects like a metal screening. The chances of tearing his esophagus or stomach or the screen getting stuck as his stomach contracts to force the metal out is much too high.
My hope he at least chewed this up a little before he swallowed as that will make passage a little easier.
You have a couple options.
One is to have him seen by his veterinarian and radiographs taken to see where the material is now. If it is in his stomach and they have an endoscope they may be able to retrieve it that way. If the screen is zinc plated this would be ideal as zinc is a toxin and can lead to stomach ulcers, bleeding, red blood cell damage, kidney and liver failure in dogs.
The other option, since he seems well so far, is to try to get this foreign material to pass through his gastrointestinal tract. And that his stomach acid helps us by dissolving the fragments.
I recommend starting with a vaseline sandwich. Take half a slice of whole wheat bread, slather with vaseline as you would peanut butter on a slice of bread, cover with the second half slice. Feed it to him bite sized piece by piece or he will make a huge vaseline mess. This adds fiber and hopefully makes the pieces slip rather than catch.
For the next few days, until the pieces pass, he'll need to be fed a bland diet with lots of fiber added in small meals several times a day, 4 to 6 meals is ideal. We do this so the pieces are surrounded, the gut isn't irritated and isn't contracting on just sharp metal.
A homemade diet for this is 1/3 boiled hamburger or white, skinless chicken, all fats and juices drained and 2/3 white rice. Add 2-4 tablespoons of canned pumpkin, (not pie filling, just pumpkin), to each meal for fiber.
You'll need to check his stools frequently for pieces in the next few days.
Signs that things aren't going well, and he needs an immediate veterinary visit, are vomiting, a tense painful belly, lack of an appetite, a fever (more than 103.5F rectally) and lethargy.
If you aren't seeing fragments in the next day or two a veterinary visit and an abdominal radiograph to see where the pieces are would be best. If it doubt at all that things aren't going well it is always best to have him checked.
Once the pieces are passed and he's feeling well then start mixing in his regular food, adding a little more regular and less bland at each meal. It should take a week to get him converted back to regular food.
The safest thing to do would be to have him examined, and try to retrieve the screen if possible, both because it is likely to be sharp and the concerns with zinc toxicity.
But he is a big fellow, and is the screen is very thin we may be able to get this to pass.
How you proceed depends upon how available emergency care is for him, and whether you feel comfortable trying things at home.
Please let me know if you have any questions.