Yes and no. It's all a matter of sophistry. Is technobabble in Star Trek magic? Are they performing rituals to have things work? People are doing nonsensical things with physical objects and spouting nonsense words - moving perspex cards round inside lit boxes, or moving fingers symbolically on glass plates full of esoteric glyphs.
Technically Doctor Who has had magic ever since they decided the main character was slightly psychic and there were species with far greater psychic powers - such as Sutekh in Pyramids of Mars or Azal in the Daemons. There are even psychic gadgets like the recent Psychic Paper. And then you have the non-technology technologies such as the Lamasteen scanner in The Lodger or quantum mnemonics: Block Transfer calculations in Logopolis and the word-science of the Carrionites in the Shakespeare Code.
Clarke's Law helps alleviate the two camps on whether there is magic in the Doctor Who universe or not by explaining it's just very advanced science we don't understand. This has been further expanded in novels and other media which go into the creation of the web of time and the first moments of the galaxy - the Time Lords set in their rules into the fabric of the universe - and the Grandfather created various backdoor bits of code that could be exploited later.
It's there - but it's mostly science. Many of the magics of the Faction are actually invokations of the Spirits - which in this case are semi-sentient eddies and currents in the Vortex that behave like the voodoo loa becuase the Grandfather and the Faction have spent a long time programming them to do so. Younger and newer members of the Faction often have faith and believe in the Spirits and the Grandfather as supernatural beings, whereas the older and more cynical members have faith they aren't and are sure they're just supranormal entities. What the belief is based on is not important to the Spirits as long as it is belief - a prime example of this is the 8th Doctor using a Faction Shrine with a nonsense rhyme1) but believing it enough to activate the correct ritual in Alien Bodies
But the short answer is - best not to explain it. The mystery is the thing that pretty much defines what magic is for. Magic thinking has been weaponised by the Faction and mostly it gets pointed at the correct target.