Car seats absolutely can be checked for free. Per united.com:
https://www.united.com/web/en-US/con...ge/infant.aspx
United accepts one stroller or one car seat free of charge as checked baggage. United is not liable for damage to strollers. Customers cannot purchase excess valuation for strollers.
Note: United will check strollers at the departure gate.
Infants traveling internationally on 10 percent of an adult fare will be granted a free baggage allowance of one stroller, one car seat and one other item (linear dimensions must not exceed 62 inches (157 cm) and weight must not exceed 50 pounds (23 kg)). If the infant’s free baggage allowance is exceeded, the excess will become part of the fare-paying adult's baggage allowance and all restrictions and excess charges will apply.
Note that the limit is one per paid passenger. For example, a solo passenger checking 2 car seats, or a stroller + a car seat would have one item go against the regular checked baggage allowance.
Note that damage can occur and many will advise against checking car seats. You can also carry to the gate and gate check, or (if your child has his/her own seat), carry it on and use it for the child. When checking in the past, we've always put it in a heavy duty canvas bag, but they do also provide those clear plastic bags at the ticketing counters. More than anything, it's to protect the car seat against dirt/grime, not physical damage.