The tube pieces could lead to a blockage or cut in his intestinal tract. It doesn't sound like he ate much of it so that is better.
Zinc in diaper ointments is toxic for dogs and can lead to red blood cell damage and stomach and intestinal ulcers, but they do need to eat and absorb a certain amount to get sick. And it takes a few days for the zinc to be digested, absorbed and affect the red cells. So his change in skin around the eye color is probably not related.
The good news is that zinc is a horrible irritant to the stomach lining so they often vomit up most of the ointment before it has time to sit in the acidic stomach environment and get absorbed. I suspect that is what is happening with him.
Also the ointment he ate is very high in fats and will likely cause some loose stool and maybe mild stomach upset, but no long term effects.
If he ate a large tub of it we might have more concerns, but if he ate part of the smaller 2 oz tube then he probably didn't eat enough of it for that to be of concern. For a big dog his size (103pounds) if he didn't eat much, you may be lucky enough not to see any diarrhea.
Given that he vomited several times the ointment was likely mixed in with the food and hard to see, but likely much of it was vomited back up.
Since he is still nauseous even though most of the ointment is probably vomited back up you can use an acid reducing medication to help with that. 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 20mg tablet per 40 to 80 pounds of body weight every 24 hours.
These will reduce stomach acid and should help settle his stomach. They are quite safe and can be used for several days if necessary.
Pick up his water for an hour or so to let the acid reducer work. Then just give him small amounts.
I would hold off feeding anything until later today or tomorrow morning, then I'd feed a bland diet. A homemade diet for this is 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. Feed small meals several times a day.
If his stools are normal for 3 days you can then transition him back to his regular diet by slowly adding more regular and less bland over 4 to 5 days.
All in all I think your fellow will be fine if we can stop his vomiting, please let me know if you have any further questions.