My answer is no by his own design.
My understanding is that he deliberately created a universe where all of his actions stay fixed, while other people have some free will. Note that my answer is based on Before Watchmen comics, and very little on Alan Moore's words.
But firstly, he does not see himself as Jon Osterman anymore. Remember the line from Watchmen #12 (emphasis respected):
It didn't kill Osterman... Did you think it would kill me?
Or from Dr. Manhattan #1 (emphasis respected):
I am not Jon, but Jon became me. His perspective informed me, his perspective--
-- created me? Is that possible? What happened in that moment, that surge of energy that resulted in my birth? If I am a quantum effect, what was the quantum cause?
Whatever it was, did it happen before the incident, or during the incident?
But my main argument comes from Before Watchmen: Doctor Manhattan.
In this prequel, there existed multiple quantum realities, each of them being a reflection of different choices people made.
It all starts Dr. Manhattan going back in time to the point Jon Osterman was destroyed in the intrinsic field chamber. Manhattan expected to see his creation, but instead Osterman left the chamber unharmed. Apparently, Dr. Manhattan had somehow caused a disturbance in the timeline, causing the reality to splinter.
Here and further, all images are from Before Watchmen: Dr. Manhattan #3
[Dr. Manhattan] The only way this could happen... The only way one quantum reality could splinter and fracture into an infinite number of possibilities with me at the center... Is if I somehow became the quantum observer.
I didn't realize it because I had only followed the stream of my existence up to a certain point, never going back to a moment before the incident the field chamber.
Until I climbed the mountain high enough to see everything around me.
At some point I did something, saw or touched something that tore this quantum reality and created another. The effect then rippled back and forth along my timeline from the moment of my re-creation, creating new realities every time I made a decision.
But how? When did I--
[Janey's words from a flashback] "Jon, I think I'd like to go home now, please."
Dr. Manhattan had altered the reality during the first meeting of Crimebusters. When he had to be paired with Rorschach, he paired himself with Silk Spectre.
[Dr. Manhattan]: What's in the box?
[Captain Metropolis]: Our first is Dr. Manhattan, whose presence here tonight honors all of us--
[DM]: It's all a matter of perspective.
[CM]: ... and the one he's partnered with tonight is--
[DM]: And perspectives can be changed... with the slightest nudge.
This choice had caused the timeline to fracture at multiple points, with Dr. Manhattan being at the centre - as he is the cause.
What does th' main man do? He goes an fixes the points in time (TM) that would've otherwise lead to nuclear Armageddon. He fixes his own choices, so that "others can still make theirs".
[Jon's father, from flashback] "... There's nothing broken so bad that you can't put it back together again with enough effort."
[Dr. Manhattan]: As I look at the random moment that brought Janey and me together, I know what I must do to fix the delicate clockwork mechanism I have broken.
Once I became the quantum observer, every time I made a choice it fractured reality, over and over. So the only way to reunite the timelines is by erasing all those choices.
The irony is not lost on me.
I am the most powerful being in the known universe. I can do anything.
But for the world to be what it needs to be, what it must be, I must sacrifice my own choices, my own free will, so that they can have choices.
He summarises it all himself in this panel:
Like my mother, I must give up all of my choices so that they can make of this reality what they wish.
In this new, united timeline, whatever I have done in the future is done. The decisions are made, and cannot be changed. they are as irrevocable as the choices I made in the past.
Because all those moments are the same moment, one moment, eternal and simultaneous.
So I must freeze that moment, that eternity, like fly frozen in amber.
I am limitless.
But I must become limited... and thus let this one world, this one reality, survive intact.
Therefore, if we count Before Watchmen continuity as canon, we have a pretty good explanation why Dr. Manhattan doesn't have free will: the universe where he doesn't have free will was the only universe where 1) he prevented nuclear war and 2) other people have free will.
To add an example from Alan Moore's comics, here's a panel from Watchmen #4, page 16:
Janey: So, what you're saying is you knew he'd get shot? Jon, I... I mean, if you're serious, I mean, why didn't you do something?
Dr. Manhattan: I can't prevent the future. To me, it's already happening.
J: Jon, what are you saying? That you know the future? About everything? About us?
DM: In 1959, I could hear you shouting, here, now, in 1963. Soon we make love...
J: Just like that? Like I'm a puppet? Jon, you know how everything in this world fits together except people. Your prediction's way off, mister.
DM: No. We make love right after Wally arrives with the earrings I ordered for you...
J: Shut up! You're messing up my mind, Jon! Sometimes I think you're messing everything up!
I mean, all this new technology, all because of you! Things are happening too fast. Things shouldn't... Was that the doorbell?
Wally: Janey? The mailman delivered this to me by mistake. Sorry I didn't drop it by earlier. Say hi to Jon for me.
Janey: Uh... Uh, sure. Thanks, Wally.
As further reading, I suggest A Timely Encounter: Dr. Manhattan and Henri Bergson by Christopher M. Drohan, Free Will and Foreknowledge: Does Jon Really Know What Laurie Will Do Next, and Can She Do Otherwise? by Arthur Ward, and I'm just a Puppet Who Can See the Strings: Dr. Manhattan as a Stoic Sage by Andrew Terjesen, all collected in Watchmen and Philosophy, edited by William Irwin and Mark D. White.