Substitute Cox, who has spoken openly about his mental health struggles including an attempt to take his own life, walked out of Ochilview during the Scottish League Two clash on Thursday night.
The 32-year-old declared himself finished with football, having previously vowed to walk off the field if taunts over his mental health continued.
Stenhousemuir referred the incident to the SFA and the governing body has confirmed a probe is under way. The investigation will be led by compliance officer Andy Phillips, who is set to speak to both players, match officials and clubs.
Cox outlined his feelings in a video posted on social media.
Sitting in a vehicle with his Rovers tracksuit on, Cox said: "The second half is just starting, Albion Rovers and Stenhousemuir. I have left the stadium.
"I wasn't playing tonight, I was on the bench, one of the boys on the Stenny team... we were having a wee bit of to-and-fro on the bench and they had a wee go at my mental health. Told me I should have done it right the first time.
"So I promised myself the next time it happened I'd walk off the park... I am done with football. Completely.
"Some folk might not think it's a big deal, I'm f*****g fed up listening to it. Don't get paid enough for it and if they'd put me on the park I'd probably have broken the boy's legs, deliberately.
"I tried to speak to the referees about it, they didn't want to know because they didn't hear it. It's the same s*** we're talking about all the time, we talk about racism and the personal issues that get brought up in football, because it's not heard by officials or whatever there's nothing they can do about it.
"So I'm going to do something about it and for me it's leaving the game. I'm done with it."
Tiffoney described reports that he attacked Cox's mental health as "simply untrue".
In a statement released through his lawyers, Bannatyne, Kirkwood, France and Co, on Friday, Tiffoney added: "Throughout the first half of the game, I was subjected to abuse and disparaging comments from the Albion Rovers dug out. I did not react to these.
"David Cox, who was a substitute and not playing, was booked by the referee for entering the pitch and abusing me. I did not abuse him. His abuse of me took place in front of the linesman. Had I made the comment, which it has been said that I made, then it is inconceivable that it would have not been heard by the linesman.
"Having been personally subjected to disparaging and hurtful remarks as a player for many years, I condone any such practice.
"I would never attack another player's mental health or wellbeing and will fully cooperate with the Scottish FA investigation into this matter."
Rovers offered Cox their "unconditional backing and support" after the "unforgiveable incident".
After speaking to both players, Stenhousemuir stated that each were making claims of "serious and wholly inappropriate comments..... that have implications for player wellbeing and mental health".
In a fresh statement on Friday, the Larbert-based club said: "Serious allegations have been made and they must be fully and robustly investigated.
"Following discussions with the Stenhousemuir player involved, he will take a leave of absence until the investigation is concluded."
A number of Cox's former clubs, including Airdrie, Forfar, Peterhead and Annan, gave him their public backing in the wake of the incident, along with other Scottish football figures such as former Celtic striker Chris Sutton.