What did the label say?

nothing on label of when to water or anything about keep dog off

What is the product?

In the absence of any additional information, most commercial lawn fertilizers should be watered in well after application. And the concern with pets is related more to any herbicides or weed killers combined with the fert (like a weed'n feed product) rather than with the fertilizer alone.

If you don't water it in NOW and wait until later, then you risk the morning dew dissolving the fertilizer slightly and making a strong solution of wet fertilizer. That strong solution can damage the plants and possibly irritate your pets' feet. If you water it in right away you dilute it a lot and wash it down to the roots.