The former world number one has faced criticism for the way he conducted himself at Augusta National.
Some were unhappy with Woods as he spat, swore and kicked his clubs to the ground during the tournament.
"It's an unedifying spectacle, there is no doubt about that," said Dawson. "Public sanctions would not be a bad thing. They would be more likely to lead to a correction of bad behaviour."
Woods finished tied for 40th on five-over-par.