If I'm playing as the Dark Side and I destroy all three of my opponents objectives do I win the game?

If you were asking if both Sides have the same Victory Conditions the answer is no.

As the Dark Side player you only have 2 ways to win the game: get the Death Star Dial to 12 or Deck your opponent.

As the Light Side player you also only have 2 ways to win: Destroy 3 Dark Side Objectives or Deck your opponent.

As the Dark Side player, when you Destroy a Light Side Objective, you increase the Death Star dial by 1 for each Objective in your Victory Pile. In other words, you destroy your first LS Objective and the Death Star Dial goes up by 1. You destroy your second LS Objective and the Death Star Dial goes up by 2, etc.

If the dial was at 6 before your Conflict phase and you had not destroyed any objectives up to that point then destroying 3 in one turn would win you the game. The dial would go up 1, 2, and 3 that turn. Other than that simply destroying them does not make the DS win.

Destroying 4 objectives as DS will almost always win the game (you'll get 10 clicks on the dial from objectives, and it will have to be at least your 2nd turn), but again that's not because of the number of objectives destroyed but because of the DS dial.