It may be difficult to answer this question using statistics. For example, the abstract to the article which rob cites in his answer says,
The most frequent accident type among collisions between cyclists and cars at bicycle crossings was a driver turning right and a bicycle coming from the driver's right along a cycle track.
This type of accident may be frequent but it ought to be avoidable (however actually avoiding them might need better education for casual cyclists, which might be difficult to achieve).
My a priori guess was that most accidents are caused by someone doing something silly (legal, perhaps, but imprudent, such as betting your life that a turning-right driver will see you or expect to find you on their right), i.e. something other than this cautious law adopted by Idaho (I call the law cautious because it still requires a yield or stop), and that therefore the effect on statistics, which aggregate all accidents for all causes, would be small. IMO my initial guess is supported by the following statements from Meggs' article:
Microfilm archives of police incident reports from 1966 to 1992 were consulted over a
period of days, and deemed too difficult to analyze; archival copies of statewide yearly summaries of
traffic injuries and fatalities, including summaries of fatalities and injuries by county and by mode, were
located instead as best available data.
Data for 1974-1975 showed that “Passed Stop Sign” was a contributing circumstance in only
5.4% and 4.2 % of injury/fatality collisions respectively in Ada County (Boise).
The amount of effort that Meggs put into research and analysis may make his article unique, i.e. the only data which we have on this subject. To answer your question I will therefore only read/critique his article: as follows.
I am inclined the discount/ignore the comparison between cities, i.e. that evidence is too weak to prove anything, because IMO there are too many variables: driving habits, available routes, qualities of the roadways, numbers of cyclists, etc. He has cherry-picked to find cities which he thinks are comparable, but he only picked two cities, so there is sampling bias and law of small numbers.
Bicycle Route Impairment
I find this a plausible mechanism, which could explain why it's safer. The law means that cyclists can choose lightly-vehicled residential roads which have many stop signs (which deter cars and prevent high-speed cars), instead of travelling on main roads.
Perhaps (there's no data) this (cyclists changing their regular routes) was a cause of the "14% drop in accidents" which he alleges happened in the year after the law was introduced.
"14% drop" a semi-impressively-large drop. It would be more convincing iff we knew the variance (standard deviation) between years; or a detailed analysis of the individual accident reports from those two years only (if "1966 to 1992" is too much data to analyze)
Stops Hurt Bicyclists and Bicycling: Safety in Numbers, Pollution, and Shifting to Arterials
I find this plausible, concurring with my own experience (except that including a "Pollution" argument is IMO an out-of-place red herring).
I don't understand what he's arguing about signals. He seems to be saying that a cyclist shouldn't attempt to communicate with other drivers, e.g. using hand-signals, which seems counter-intuitive and contrary to my experience; however he cites supporting literature for his points, for what that's worth.
I disagree. There may be some advantage to setting off first, e.g. just before your light turns green when there's no cross-traffic; but that may increase risk from cross-traffic running a yellow. If he wants to make that argument, there are safer solutions than allowing people to jump off before green light, e.g. the use of bike boxes.
Reading indirect reports of the cited "A 2007 report by Transport for London" suggest that many of "accidents affecting women" are caused by their being next to lorries turning left (equivalent to "turning right" in American right-hand-side roads), which is an IMO too dangerous place for a cyclist to be in the first place: instead of jumping the light, I would get in lane in front of or behind the truck.
In summary, IMO this part of the argument is weak.
Predictability Reduces Risk
Again the supporting data is a one city to one city comparison, which I find very weak.
Injurious Falls at Stops
I disagree: the cyclist must be ready to stop, even if (in Idaho) they end up deciding they don't need to.
Everyone falls when they first learn to use toe-clips (or "clipless" cleats): the solution is to practice away from traffic until you know how to use that equipment, or, don't use that equipment if you can't use it safely. Inability to come to a stop without falling over isn't fixed by letting people run stop signs.
There's also a difference between de facto and de jure: in that a cyclist can obey the spirit if not the letter of the law, by coming to "virtual" stop of about one mile per hour, without unclipping but being ready to unclip if necessary, before proceeding. anecdote
The usual fix is to ride on roads with fewer stops, which are more arterial. I'd guess that's to go faster rather than to avoid injury. It's obviously true that you can't go as fast if you keep stopping, except with much more effort and therefore injury; but I'm not sure this argument ("avoiding injury") would change his opponents' opinions.
Criminal Antagonism: Cultural Deterrents
I find that plausible; when a Toronto paper publishes an article about cycling, my impression/memory is that many or most of people's complaints about cyclists (in their comments to the article) are about cyclists lack of stopping: through jealousy perhaps ("I have to stop in my car, they should stop too"). Other IMO fewer complaints include "ninja cyclists" wearing black with no bike lights, and cycling on the sidewalks.
My chief hypothesis is therefore that letting cyclists roll through stop signs would make it more likely that cyclists wouldn't so much avoid residential roads with stop signs. Doing this could make those cyclists safer.
On the other hand, cyclists still need to be able to yield at stop signs, which they can't if they don't slow down from their 20+ mph cruising speed, so they might still choose the same arterial roads as cars.
A way to test this hypothesis would have been to see whether introducing the Idaho law caused cyclists to change their routes. I guess that survey hasn't been done.
If that is the primary reason, then there may be other more effective things that legislators could do: for example bike lanes along arterial roads, sharrows, bike boxes, and multi-use paths.
A secondary hypothesis is that having a different law for cyclists might encourage more cycling (eventually resulting in increased safety after new cyclists become experienced and drivers get more accustomed to watching out for cyclists); and possibly reduce anger which drivers feel towards cyclists, who are mostly scoff-laws who bend and break the existing "come to a complete stop" laws.
I note that he seems to be cyclist who is advocating for the law, so I expect he would have some personal bias.
As a final note I'd suggest that a law which restricts people's actions shouldn't exist without good reason. If that's so then perhaps the question shouldn't be, "Does not-stopping make cycling safer?", but rather, "Does not-stopping make cycling more dangerous?" The Idaho law still puts a burden of responsibility on the cyclist: to stop and/or to be willing to yield, to proceed only when safe, and (laws of physics) to be personally injured or killed if they're careless.