Well, we can look back in history to see what such things were like. I'll use the murder rate as the proxy for gun violence.
Prior to 1968 there were no federal controls on most firearms (excepting full auto, suppressed, short barreled, etc). They were advertised in comic books. Kids could buy them in many states. Most states had no background checks. What was the violence rate? We're just getting back down to the rates of the 1960s, and that was much higher than the 1950s.
Prior to 1938, there were no federal restrictions on purchasing any firearms. Even Sears sold full auto guns. Crime peaked during Prohibition, never to go down to pre-Prohibition levels again.
This pattern is true in other countries, too. Here's what a British "bobby" found out when he did some research:
"No matter how one approaches the figures, one is forced to the rather startling conclusion that the use of firearms in crime was very much less when there were no controls of any sort and when anyone, convicted criminal or lunatic, could buy any type of firearm without restriction. Half a century of strict controls on pistols has ended, perversely, with a far greater use of this weapon in crime than ever before." - Inspector Colin Greenwood, _Firearms Control_, (Routledge and Keegan, London, 1972) p. 243
"At first glance it may seem odd or even perverse to suggest that statutory controls on the private ownership of firearms are irrelevant to the problem of armed crime, yet that is precisely what the evidence shows. Armed crime and violent crime are products of ethnic and social factors unrelated to the availability of a particular type of weapon. The numbers of firearms required to satisfy the 'crime' market is minute, and these are supplied no matter what controls are instituted. Controls have had serious effects on legitimate users of firearms, but there is no case, either in the history of this country or in the experience of other countries in which controls can be shown to have restricted the flow of weapons to criminals or in any way reduced armed crime." - Inspector Colin Greenwood, "Shooting Back," Police Review, 10 November 1978 page 1668
America's foremost researcher on gun violence also feels about the same:
"...Consequently, when medical journal authors report that there is little evidence on a given topic, it may often really mean only that they made no serious effort to find any or chose not to report what they found. For example, in an article published in 1996 in the Journal of the American Medical Association, Douglas Weil (research director of the Center to Prevent Handgun Violence, affiliated with Handgun Control) and a colleague claimed that "there is little published research on the effectiveness of gun laws" (Weil and Knox 1996:60). In fact, there were, at the time this article was published, at least forty-five empirical studies of the impact of gun laws on violent crime, suicide, and gun accidents (Tables 8.4 and 11.1). Weil then proceeded to inaccurately claim that "with little dissent, these studies are generally supportive of the thesis that well-tailored gun laws can have a beneficial impact" (ibid.:60), when in fact the studies have generally indicated that gun laws, whether "well-tailored" or not, have no measurable impact on violence rates (Chapter 11; PB;Chapter 10)...." Page 42, Gary Kleck, _Targeting Guns_, (Aldine de Gruyter, NY, 1997)
Others have pointed out that the criminals are willing to break the law to further their criminal activities. Increasing legal availability isn't going to increase the "illegal" availability very much, if any.