Mullin left the Snapdragon Stadium pitch in an oxygen mask after being wiped out by the goalkeeper in Tuesday's 3-1 friendly win in San Diego.
Mullin, who banged in 47 goals last season, beat Bishop to a lofted through ball on the edge of the area in the 12th minute before being whacked into the chest.
Medics instantly rushed to his aid, with the ex-Cambridge marksman requiring treatment for seven minutes along with an oxygen mask before being substituted.
After staggering off the pitch, Mullin was then taken to a nearby hospital where he was diagnosed with a punctured lung.
He later took to his Instagram story to provide a positive update on his health, where he shared a picture of himself giving the thumbs-up while laying in a hospital bed.
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Captioning the post, he said: "Thank you for all your messages! All good here!"
Bishop only received a yellow card for the reckless challenge, with Wrexham boss Phil Parkinson demanding he should have been sent off in his post-match interview.
The 23-year-old was public enemy No.1 in the eyes of the United fans present at the stadium, who booed the ex-Southend shot-stopper with every touch.
He was eventually replaced by Radek Vitek for the second half, with supporters even jeering the teenager after mistaking him for Bishop.
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Bishop later issued an apology on social media, where he tweeted: "Just wanted to share my sincerest apologies to @PMullin7.
"A complete misjudgement and a genuine accident with malicious intent at all!
"Wishing you the speediest recovery and hope to see you back scoring goals as soon as possible!"
United coach United coach Travis Binnion, who took charge of the youthful side that lost to the National League champions, defended Bishop after the match.
He said: "Obviously, there's a challenge in the first half when Mullin gets injured, we hope he's alright.
"I know Bish has reached out to him already and the referee could have sent him off for that.
"It's part and parcel of the game, nobody wants to see anyone get injured. He's made an honest challenge, he's come off worse. He's already reached out to him.
"Booing him, it is what it is. He's tried to reach out. He's a great lad Bish, he hasn't got a bad bone in his body."
Meanwhile Wrexham goalkeeper Ben Foster told The United Stand: " I feel so bad for Nathan.
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"He's absolutely heartbroken, honestly, he's absolutely heartbroken, I passed Paul Mullin's number on to Nathan so he can get in touch with him.
"But he's absolutely gutted, I feel for the lad."