Can A Dog Rescue Repossess My Dog?

I adopted a dog about 4 months ago who just turned 6 months old. The rescue I got him from is saying he must be neutered now. I want to delay the neuter until he’s 1 year old because of the overwhelming orthopedic health benefits. I 100% plan to neuter him and even have it scheduled at his vet. Another note, they told me I could use the vet I’ve been using for the last 17 years due to their vet being so far away from me. The vet who I have been using has advised me to wait until one year old. They told me I agreed in the contract to neuter at 6 months but I can’t find that anywhere. Basically, I just want to know if they can take the dog from me. That would absolutely devastate my kids. Here is the contract:

I, (my name), understand and hereby acknowledge that I am adopting the above described animal hereinafter referred to as 'Animal", whom has not yet been sterilized. I represent and attend that I understand and acknowledge that Georgia State Law HB 1181, effective 1 July 1994 requires that any dog or cat adopted from a public or a private animal shelter, refuse, or humane society must be sterilized. This addendum shall be considered part of and incorporated into the (rescue’s name) Adoption contract. Per the contract, any adopted animal must be sterilized pursuant to HB 1181. I understand that I will not incur additional cost for the sterilization procedure, which is already covered by the adoption fee paid to (rescue’s name) if the Animal undergoes the surgery at a veterinary clinic which has been approved and pre-arranged by (rescue’s name). I acknowledge (rescue’s name) agrees that it is in the best interest of the Animal to be placed into its permanent home as a foster until such time that the Animal is physically able to undergo the procedure and the veterinary clinic has the availability to perform the procedure.I further agree to keep in weekly contact with representatives of (rescue’s name) regarding any issues the Animal may have. I understand and acknowledge that I legally must transport the Animal to be sterilized to the designated (rescue’s name) Veterinary Clinic within 1 week of receiving confirmation from (rescue’s name) that the Animal is ready to undergo the sterilization procedure by the treating Veterinarian. Furthermore, I understand and acknowledge that not complying with the sterilization as described in this addendum will result in the breach of contract and consequences thereof referenced in the (rescue’s name) Adoption Contract.