Probably. Palpatine was the only one required to determine that the conditions necessary to trigger Order 66 had been met; having made that determination, he was the only one legally empowered to issue the order.
Order 66: In the event of Jedi officers acting against the interests of the Republic, and after receiving specific orders verified as coming directly from the Supreme Commander (Chancellor), GAR commanders will remove those officers by lethal force, and command of the GAR will revert to the Supreme Commander (Chancellor) until a new command structure is established.
Republic Commando: True Colors, by Karen Traviss (Wookieepedia)
It's Legends canon, but let's take this as the wording of Order 66.
First, does the Jedi Order have the legal power invested in them for Mace Windu to do what he did, or is it considered extralegal? Based on the legal definition of Order 66, does Mace's actions fulfill the conditions necessary to legally invoke it?
And does Palpatine have the executive power to unilaterally invoke Order 66 without consulting or getting approval from any entity?
The Wookieepedia page has some discussion on this:
- The Supreme Commander is the only one who can issue Order 66. Palpatine is the Supreme Commander.
- There is no requirement for anyone other than the Supreme Commander to make the determination that the Jedi are acting against the Republic.
- There is no requirement for the order to be authenticated with any other command authority e.g. GAR High Command.
As soon as the Jedi led by Mace show up at the Chancellor's office with the intent to arrest him, it seems to me that the Chancellor can take the view that that is an attempted coup d'etat 'against the interests of the Republic' and from that point onward he has all the legal cover he needs.