Norris had looked unwell all weekend and said on Saturday that he had been "suffering with my throat and my eyes and seeing", saying it was something other than COVID-19.
On the grid before Sunday's race, Norris continued to look out of sorts, but once the visor went down the Briton produced a solid race to finish eighth, with teammate Daniel Ricciardo down in 12th.
After a sweltering race Norris cancelled his media duties to focus on his recovery, and team principal Seidl later explained that his driver was struck by tonsillitis.
"Obviously you're not feeling well all weekend when your energy levels being low with the tonsillitis he was suffering from," Seidl said.
"It was a very challenging race, especially in these hot temperatures, so he was pretty flat at the end of the race and after the race but thanks to the great medical support within the team that he has around him.
"The objective is clear now the next four days, we need to make sure he gets some good rest now in the next four days before we then go again in Monaco."
Seidl praised Norris for fighting through his illness to bag points for the team on an afternoon where ambient temperatures soared to 36 degrees, but only after the team's medical staff gave him the green light to start.
"I can't thank him enough for battling through and he showed a great fighting spirit because in the end it enabled us to score these four points today," Seidl added.
"It was just great to see the commitment from his side but at the same time it's also clear we obviously have a doctor involved in decisions like that and we would never risk the health of Lando, just to be clear on that as well.
"But after the doctor gave the green light today and after it was also clear from Lando's side that he would give it a go, we went for it."
If Norris cannot start in Monaco next week, the team would have Mercedes drivers Nyck de Vries and Stoffel Vandoorne available, with Alpine reserve Oscar Piastri also in the Woking team's reserve pool after a deal between the two teams was struck in March.
Seidl would not be drawn on any potential replacements, saying the team would make a decision at short notice if needed.