Welcome to Pigeon Angels, Ohons.
For whatever reason (e.g. too young to fly, ill or possibly a preexisting injury), the dove was grounded and therefore allowing the cat to catch him.
The good thing is the cat brought the dove to you.
In what general location do you live? The reason I'm asking is a rehabber or vet, who treats pigeons, might be nearby who could assist you.
You can certainly call your vet and ask if they treat doves and if not, perhaps they know of a rehabber who can.
You are exacly right, in that you don't want to put him back outside.
If you have a small carrier or even a box, that you could line with a towel, to put the dove in would be good. It's best to keep them quite and away from family and/or animal traffic.
Is there any chance of posting a picture of the dove and the injured wing?
A Pigeon's Prayer
Please watch over us while we fly,
Keeping us safe from the predators that share the sky.
If we become ill or injured in any way,
Please lead us to safety where we are welcome to stay.