The 26-year-old has sealed a three-year deal with the Hoops after completing his £1million switch from the Hawthorns.
Now the 47-cap Costa Rica international is looking to kick-start his career after a frustrating spell with his former midlands employers saw Pulis hand him just three appearances over the past 12 months.
However, the former Rosenborg and Copenhagen defender insists his confidence levels have not been dented by his stint on the sidelines.
Speaking as Croatian defender Jozo Simunovic was heading towards the Parkhead exit after being lined up for a move to Torino, Gamboa said: “I have nothing to prove. I was never given a fair chance where I was.
“But now I think I will get it and I will fight for it.
“It was difficult there because sometimes it doesn’t matter what you do on the pitch because the other situations around don’t help.
“But I always kept my head on what I wanted - to be fit, with no injuries - and I knew my chance would come. And what a chance I’ve got here now at Celtic.”
Gamboa will now get the chance to line up for Brendan Rodger’s team in the Champions League.
But he stressed he would have been keen to join up with the Scottish title winners even if they did not have glamour games with Barcelona, Manchester City and Borussia Monchengladbach to look forward to.
He said: “When my agent called me to tell me about Celtic I said to him, ‘Do everything because I really want to go’.
“I know everything about Celtic from back home. I know Costa Rica is far away but you hear about Celtic a lot there, so I was really excited when I was told they were interested.
“Now we are here and I’m really happy.
“I spoke to James Morrison, Darren Fletcher and Graham Dorrans who were with me at West Brom and they said I should jump at the chance.
“It doesn’t matter if they play Champions League or not - it’s still a big club.
“So for me the fact they have qualified is like a double bonus - I get to play for this club and to play in the Champions League.”
The Latin American was signed by Rosenborg in 2012 as the successor at right-back to Mikael Lustig after he made his move to Celtic.
Now Gamboa has said he is ready to replace both the Swede and former former Manchester United youngster Saidy Janko in Rodger’s team.
Asked if he was ready to feature after such a long spell out of the picture with the Baggies, he said: “I always try to keep myself in shape by doing extra work. I wasn’t playing at West Brom but I was playing in all the national team games.
“So right now I am missing some match fitness but I will come here and do my best and if you give me the chance I’ll be ready.
“Wherever you go, you will have to fight for your place. It doesn’t matter where you go, you have to earn your chance.
“So I am ready to fight and to put pressure on the other right-backs - but the gaffer will decide everything.
“I remember when he was coaching Liverpool and the way he likes to play. So I know I will have a chance to fit in and do my best.”