No. LED headlights will not damage your car if properly installed. In addition, you want to purchase the correct LED light that will fit your car. For example, if your vehicle has a CANbus system, you want to buy the right product and install it correctly to avoid any potential issues.
If your car was not originally designed for LED headlights, you can purchase an LED headlight conversion kit. They are safe and will not damage your vehicle as long as you buy top-quality products.