Booting a road bike tire I ran over a thorn which went through the tire and punctured the inner tube. The thorn left a small hole of about 1-2mm in the center of the tire.
The tire is a pretty new Specialized turbo.
Considering I ride with 100+ PSI, will it suffice to just fix/replace the inner tube or will I have to boot the tire?
Thorns generally don't make neat round holes as per your picture.  While its not impossible, this looks more like a wear indicator, a hemispherical pit on the tyre's tread, intended to show when the tyre needs replacement by wearing off.
If you can hold the tyre up to the light (when its off the rim!) and see light through, then your inner tube could herniate out and will wear another puncture in short order - perhaps 100-200 km of riding.
If you can't see the hole on the inside, its not worth worrying about.  There may be more than one wear-indicator pip on your tyre too, so look around.  Also check the other wheel, if you have the same brand/model of tyre on both ends.