This hedges awfully close to politics and is sort of tangential, but nonetheless...

The argument (stronger gun restrictions lead to less violence) to which you are referring is predicated upon two things: the perception that the restrictive gun laws of Japan and the UK are causally linked to the low incidence of gun violence, and that gun violence is in some qualitative way worse than other channels of violence; e.g., that dying from a stab wound or a bomb is in some way not as bad as dying from being shot. (There's some interesting stuff to explore in terms of human perceptions of risk and such in that, come to think of it, similar to the perception of safety in an airline vs. in a car -- something about control, perhaps?)

In any event, there may be a causal relationship between the restrictiveness of gun laws and the incidence of violent crime (not just gun violence, but violent crime in general); I don't have the references to hand, but there have been some studies that have found some statistically significant correlations that actually controlled properly for confounding factors (ISTR one looking at Philadelphia, but even that one was imperfect). Not only that, but while there are US states with liberal (i.e. low-restriction) gun laws and high violent crime rates, there are also U.S. states with liberal gun laws and low violent crime rates (Vermont has almost no gun laws to speak of), and there are U.S. states with long-standing extensive gun regulation and perennially high violent crime rates.

Furthermore, there are countries that have restrictive gun laws and very high crime rates (Russia's at the top of that list, AIUI), and still other countries with gun regulation more restrictive than most of the U.S. but far more liberal than the UK or Japan, but low violent crime rates. In rural Finland, for example, it's not unusual to see people walking around with semi-automatic AK-47-type rifles slung on their back, and nobody thinks anything of it, because such firepower is sometimes necessary to deal with particularly ornery wildlife. (Handguns are heavily stigmatized there, mind, and very tightly controlled.)

There are many other factors that are much more strongly correlated with, and probably causally related to, violent crime in general and gun violence in particular than the restrictiveness or liberality of firearm legislation, such as poverty or corruption. While there's psychological and political reasons why the discussion around reducing violent crime in the U.S. frequently revolves around guns and gun laws, I've long been of the opinion that passing new gun laws while neglecting social welfare is at best inefficient, given the costs of enforcement, and most likely completely ineffective.