My Opinion, I am not a phD or Professional grower, so it is what it is.
Deionized is (sorry German here) as far as I can tell basically distilled water or? basically that is water with 0 PPM. This usually will have a PH of 7.0 (anything else indicates that it is not very pure, so I'd leave that one brand alone. Basically this is much the same as using a Reverse Osmosis Filter (which I would recommend / prefer).
Short answer is: yes you could, but make sure to add your foods and remember that in most water you have trace elements like Iron etc. In Deionized water you do not (or should not) so you may find your feeding changes a little and you have to add Cal/ Mag / Iron etc but honestly, I am on week 12 of a grow only using PH Neutral water and occasional tests and watching the plant to see what is missing or in abundance. The only issue I had was forgetting to add the right food for 3 days...
And once you added everything, re-PH the water for your grow medium (6.5+- for Earth? sorry, I only grow in Coco, but I think that's about right). Just get a Growers PH Down, you wont need Up. Good luck and again, I really recommend buying an aquarium style R.O Filter (or even an inline filter), it WILL save you in the long run if your water is that bad.