The striker, who spent the first half of the campaign in League One with Fleetwood, completed a temporary switch to London Road yesterday.
Morton scored seven goals in 22 games for the Cod Army this season, having previously enjoyed loan spells with Lincoln, Northampton and Braintree.
Posh owner Darragh MacAnthony said on social media he signed the 21-year-old with a view of making the move permanent in the summer.
Morton, though, said the move is about proving to Albion he can cut it in the Championship.
“I need to prove to people at West Brom I can do it at this level with the aim of going back there and playing,” he said.
“Of course, my focus is on the here and now and I want to score goals and influence games.
“Once the move came about it was definitely something I was interested in.
“It’s the next step for me, I need to test myself in the Championship.”
Last season, Morton netted three goals in 21 games during an injury-hit spell with Lincoln.
The striker says he hasn’t scored as many goals as he would have liked in his last two loan spells.
But he netted eight in 12 games for Northampton.
And he is confident he can score regularly in English football’s second tier.
“I think I’ve added something different to my game from each of my loan spells,” he said.
“Maybe the goals haven’t come as prolifically as I would have liked in the last couple of loans.
“But hopefully the goals will come here.
“I think in the right system, like I showed at Northampton, I think I can score goals and I think this move will definitely suit me.”
Meanwhile, Albion have confirmed new recruit Daryl Dike will wear the number 12 shirt this season.
The number had been worn by Matheus Pereira before his move to Al Hilal.