I'm gonna restate this in terms of Rule of Law and discarding Basking Rootwalla to Wild Mongrel, so that we can answer the question.
Let's say you control Wild Mongrel, and Rule of Law is in play. You discard Basking Rootwalla to Wild Mongrel, and you remove it from the game. The "When this card is removed from the game this way, its owner may play it by paying [cost] rather than paying its mana cost. If that player doesn't, he or she puts this card into his or her graveyard." trigger goes on the stack, and your opponent plays Stifle, countering it.
Play: The act of playing a spell, land, or ability involves announcing the action and taking the necessary steps to complete it. Playing a spell or activated ability requires paying any costs and choosing any required modes and/or targets. See rule 409, "Playing Spells and Activated Abilities." Playing a land simply requires choosing a land card from the hand and putting it into play. See rule 212.6, "Lands." Playing a mana ability requires paying any costs, then immediately resolving the ability. See rule 411, "Playing Mana Abilities." Triggered abilities and static abilities aren't played-they happen automatically. See rule 410, "Handling Triggered Abilities."
If you didn't do that (which you didn't in this case), you didn't play a spell for Rule of Law.