It may be worth adding that the WHO sets a somewhat higher threshold for toxicity than the EPA, for chlorine dioxide in water, namely 2mg/L.
Anyhow, the problem is that there don't seem to be even animal studies finding that at non-toxic levels (for the host) chlorine dioxide helps with an already established viral infection. So basically there's zero evidence, even from animal studies that it could be a treatment.
Edit: Actually, this is not entirely correct, Peter's answer points to a [not yet formally peer-reviewed/published] preprint of a study in chicken embryos, but it's a bit disconcerting that such a study was only done recently , when chlorine dioxide has been used as a disinfectant for decades. So, you could at least say that for internal use, this is a very experimental treatment, in preclinical stage at best. Also that study noted that "developmental abnormalities were observed in the majority of the infected
embryos, including those that received ClO2 treatment", so it seems it's not that foolproof of a treatment, although treatment lowered the servility of some symptoms, including mortality. Viral load was only lowered 2.4-fold in the treatment groups (30-300 ppm ClO2) compared to controls; in absolute terms "average viral load
of ClO2-treated chicks was 104.3/mL, range: 103.66 – 105.03 and of untreated chicks was 104.83/mL,
range: 104.52 – 105.01, respectively (Tukey HSD, Group E vs. F, p = 0.03)." The study was also limited to a single dose of ClO2, which limited both toxicity [to the host] and effectiveness against the virus. Before any human trials, more complex animal models would have to be tested. E.g. look at how
oseltamivir was tested in various animal models; an antiviral like that completely inhibited virus replication
in some organs but only partially in other organs in the same animal etc. Chicken embryos are really on the lowest step of the ladder of in vivo (animal) models.
On the interesting, but not-really-relevant side, besides the plethora of studies on chlorine dioxide as an additive in water treatment (remember again, the dose makes the poison) which mostly what the OP linked to, there is also one study on chlorine dioxide killing influenza viruses aerosols. (Again this was at host-safe levels.)
Mice in semi-closed cages were exposed to aerosols of influenza A virus (1 LD50) and ClO2 gas (0.03 p.p.m.) simultaneously for 15 min. Three days after exposure, pulmonary virus titre (TCID50) was 102.6±1.5 in five mice treated with ClO2, whilst it was 106.7±0.2 in five mice that had not been treated (P=0.003). Cumulative mortality after 16 days was 0/10 mice treated with ClO2 and 7/10 mice that had not been treated (P=0.002).
Actually, part of study that strongly indicates that the relevant effect happens outside the body: if application of the ClO2 gas is delayed relative to the virus aerosol, the mice still die aplenty:
When ClO2 gas was delivered 5 min after the delivery of virus aerosols (5 min delay), mortality was 10 % (1/10) (P=0.081 versus no-ClO2 group). The mortality rate was 50 % (5/10) with a 15 min delay, which was [statistically] the same as in animals that received no ClO2 gas treatment.
Thus the ClO2 gas doesn't really treat already infected mice. (It's a bit more complicated why the window is 15 mins, basically the virus has to make its way through mucus etc.)
So, this in aerosol form it could be potentially be a more useful application against a mainly airborne pathogen, but there are substantial practical difficulties putting it in widespread practice since you really need to kill/inactivate the virus before it enters the body with this method/substance.
Additinally, the WHO and the health ministry of Japan have said in a statement that fumigation/spraying devices that are loosely based on that idea (e.g. blow a mist/aerosol of chlorine dioxide in people's faces) aren't approved or effective and that some adverse events have been reported in Japan after exposure to such devices.
(There's no technical detail in that newspaper report why the authorities cited say that, but it's probably related to the difficulty of ensuring the (effective but non-toxic) concentration in a larger volume of air from a point source etc.)
Since I see there's some discussion under the other answer about disinfectant mouthwashes...
Various kinds of these have been studied with respect to reducing the amount of virus emitted
by Covid-19 patients, particularly in a dentistry setting. (There is actually mouhtwash based on
chlorine dioxide, although it's not commonly used in dentistry.) But the effect of any such mouthwash products is quite transient, on the order of a few of hours as measured by salivary viral loads (which don't quite drop to zero). Also there has been substantial heterogeneity in the results reported, even for the same active substance; some studies found no reduction, others found a load reduction e.g. in half the patients tested, but not in all of them some reason (which is not really understood). So even for this limited purpose of acting like a "chemical mask" reducing the amount of virus emitted, the evidence regarding disinfectant mouthwashes is mixed.