Toothpaste is more likely to irritate your skin than completely heal your acne. It's true that toothpaste has a few ingredients that can help treat pimples (e.g., baking soda and hydrogen peroxide), but the risk of skin irritation, sensitivity, and redness is much higher than using a traditional acne treatment.
An alternative to toothpaste is a DIY face mask made out of baking soda. More effective treatments for acne are over-the-counter facial products that contain salicylic acid or benzoyl peroxide. Whitening toothpaste is particularly irritating to your skin; you may experience burning and skin peeling if you apply it topically to a blemish.