Can I caulk around this tub?
You certainly can, if you do it carefully. Here are my tips:


Use clear silicone. Any color (including white) will be much more conspicuous over the grout joints. Clear will transfer the existing colors and blend much better, almost disappearing.
Run a roll of masking tape along the floor, tight to the tub. Once you've tooled your caulk you can remove this for a clean line. 
Cut the tip of the tube small. You want it to fit the smaller areas of the gap well to minimize squeeze-out. At the wider areas, slow your slide rate to allow the gap to be filled well. 
Inject a substantial amount of caulk into the gap. You don't want a wall of caulk that will collapse. You want a solid rope. 
Be sure the gap is completely dry when you apply the caulk. Silicone doesn't grab wet surfaces. 
Tool the caulk with a finger, wiping the finger clean on a rag often. You don't want excess caulk oozing out around the sides. 


Good luck.