This only holds if every other variable has been controlled for. I think it's time for a thought experiment.
Let's say that the government of Ontario is interested in reducing health care costs. They have a hypothesis if there are more smokers, there will be more people diagnosed with lung cancer. So they look at the data and find that, while the number of smokers in the province has been decreasing steadily, the number of people diagnosed with lung cancer has been increasing. According to your logic, that means that the number of smokers does not cause an increase in the number of people diagnosed with lung cancer. But what if what actually happened is that people started getting tested more frequently for lung cancer, or that there was an improvement in the tests that detect lung cancer, so the numbers were going to rise anyway? Unless you control for other variables, it's really hard to make a judgment call.
Now, in the hypothetical situation where you only have X (gun control) and Y (violent crime rates) changing, and there are no Z (population), W (economy), A (political climate), D (weather), F (wealth disparity), P (inflation), Q (gun availability in nearby states), or T (number of police in the neighbourhood) factors fluctuating to complicate things, then, and only then, can you say that X and Y are in fact negatively correlated, and that an increase in X does not cause an increase in Y.
The point I'm getting at is that things are more complicated than the simple independent-dependent model that you seem to be pushing.