After you have done the quest Kidnapped, you will get a quest called A Lost Romance, if you told him to dump Linda. Go back to the Bowerstone Shelter and Orphanage a few days later and you should find him standing outside.
If you can't find him outside, he might be in The Riveter's Rest, the local pub, or walking around Bowerstone Industrial. If you still can't find him it is likely that a glitch has occured and hence you won't be able to complete the quest. Also note that Kidnapped and A Lost Romance must be completed before you leave for Aurora.
Once you have started A Lost Romance, Elliot will ask you to propose to him. If you have a wedding ring on you, you can propose straight away by interacting with him and choosing the option that says Propose (RB). If you don't have a ring, either take Elliot by the hand (LT) and go looking for one, or leave him and find one on your own.