Anyhow, tl; dr: Canada is most certainly independent, and gained its independence peacefully (although there were minor conflicts prior to that).
The Crown is the head of our State; all authority derives from our Sovereign.
http://www.parl.gc.ca/About/House/compendium/web-content/c_d...
Ultimately, all legislation requires Royal Assent, and all executive powers are granted by the authority of the Sovereign, who may revoke them at their will and so force an election.
Would you say UK is not _actually_ independent because it's a constitutional monarchy? If not so, why would Canada be any different?
http://www.parl.gc.ca/About/House/compendium/web-content/c_d...
Yes, Canada is a monarchy, yes, the Queen has the ultimate power in theory, but the head of state is the Queen of Canada, not the Queen of the UK.
I'll repeat myself, them being both the same person is incidental, if you think the fact Canada being a monarchy means it's not independent, does that also mean the UK is not?