Not a cure
This certainly doesn't apply to Type 1.
All food gets converted into carbohydrates (glucose) in the body, but some foods take longer than others. Insulin is needed to metabolise the resulting glucose into energy that the body can use. Type 1 Diabetics don't produce any insulin, so blood sugar goes up to dangerous levels regardless of what's being eaten. Refined carbs will increase blood sugar quicker than proteins and fats, but it all has the same effect eventually. Injecting insulin is the only way to manage Type 1 diabetes. Low blood sugar is what happens when too much insulin is injected, as Mike Bell has pointed out rather vividly.
Type 1 diabetics on a basal bolus insulin regime (one or two slow-acting insulin doses per day, with fast-acting insulin injected at mealtimes) can benefit from fewer insulin injections if carbohydrates are avoided. The basal dose works over a 12 or 24 hour period, so covers low GI foods like vegetables and meats. For example, a fried breakfast (without hash browns or toast) needs little or no insulin because the basal dose can deal with the slower release of glucose.