The quote titling this post - "A strict and lifelong gluten-free diet can help recover vitamin D level without any supplementation" - comes from the results published by Fabiana Zingone & Carolina Ciacci .
These authors inspected the peer-reviewed literature looking at vitamin D levels in relation to the archetypal 'diet can affect health' autoimmune condition that is coeliac (celiac) disease. They looked at both something called calcifediol, otherwise known as 25-hydroxyvitamin D (25(OH)D), and also something called calcitriol, also known as 1,25-dihydroxyvitamin D3. The difference between the two 'forms' of vitamin D is that one represents the 'pre-hormone' version - (25(OH)D) - that is typically quantified to give a 'where you're at' measurement of biological vitamin D levels, and the other - 1,25-dihydroxyvitamin D3 - is the 'active' metabolite. Through the wonders of something called mass spectrometry, these and other 'forms' of vitamin D are able to be quite accurately measured  in various biological fluids.
Zingone & Ciacci reported that "most of the studies on vitamin D in adult CeD [coeliac disease] report a 25 (OH) vitamin D deficiency at diagnosis that disappears when the patient goes on a gluten-free diet, independently of any supplementation." This finding really intrigued me. It got me thinking of a few things; notably about mechanisms and biological relationships and indeed, how applicable such a finding could be to the general 'not coeliac' population or those 'around the diagnostic edges' of CeD.
OK, it should be noted that part-and-parcel of CeD is an issue with malabsorption, where 'the body does not fully absorb nutrients' (see here). The inflammatory processes at work in CeD do some pretty awful things to the structure and functioning of the mucosal lining of the bowel which aren't really conducive to optimal absorption of nutrients from food. The implementation of a gluten-free diet does help matters; and so logic dictates that absorption of something like vitamin D from food sources will be improved when a gluten-free diet is implemented. I've also blogged about how this process *may* also be part of the effect noted in relation to [some] autism and the use of similar dietary intervention strategies (see here). Alongside, I'll mention that, done right, a gluten-free diet is actually not the most unhealthiest diet in the world either.
Having already sort-of mentioned something like non-coeliac gluten/wheat sensitivity in the contest of the 'diagnostic edges' of CeD, I do wonder if there could be a further plan of research there too. Y'know to look at questions like whether such not-quite-coeliac-disease conditions also (a) manifest as having low levels of vitamin D when not diet treated and (b) whether the use of a gluten-free or other diet (see here) might similarly positively effect vitamin D levels in those circumstances?
There's also another potentially important explanation to consider which was tweeted by Dr Emily Deans (she of the fabulous Evolutionary Psychiatry blogs): could vitamin D deficiency be a marker of something more general? To quote from Dr Deans' tweet (shown pictured above): "Because it’s [vitamin D] an inverse acute phase reactant and goes down with illness and up with health." There is some sound logic behind such an observation insofar as vitamin D insufficiency/deficiency being associated with all manner of physical and behavioural/psychiatric diagnoses/conditions/states (see here and see here for examples). Correction of vitamin D deficiency is all well and good when it comes to correcting biological measures, but outside of treating something like rickets, the evidence for extra-skeletal effects from such supplementation is currently not that great.
Finally, I'll draw your attention back to some work suggesting that vitamin D might itself have some interesting effects on gut barrier function (at least in mice) (see here). One wonders what this could also mean for CeD and the spectrum of gluten-related issues?
 Zingone F. & Ciacci C. The value and significance of 25(OH) and 1,25(OH) vitamin D serum levels in adult coeliac patients: A review of the literature. Dig Liver Dis. 2018 Apr 13. pii: S1590-8658(18)30702-3.
 van den Ouweland JM. et al. Vitamin D and metabolites measurement by tandem mass spectrometry. Rev Endocr Metab Disord. 2013 Jun;14(2):159-84.