The answer is yes. 

Look no further than fighters such as Evander "The Real Deal" Holyfield who tore his left rotator cuff in the second round in his first fight against Michael Moorer and was able to fight a full twelve rounds while throwing hard punches and some flurries throughout the entire fight. Holyfield just barely lost a decision to Moorer.

Andre Berto also tore his main supporting tendon in his right rotator cuff during his fight with Jesus Soto Karass and was able to fight 11.5 rounds before getting knocked out. 

Another example is Vitali Klitschko, who tore his rotator cuff in the third round in his fight against Chris Byrd and was ahead on points at the time he quit on his stool after round nine. Had Klitschko simply fought three rounds against the light-punching Byrd without using the injured shoulder, he would have retained the WBO heavyweight title. 

Lastly, another example is Jesus Chavez who fought Erik Morales with a torn right rotator cuff which he claimed he tore in the second round but fought valiantly for a full twelve rounds. 

There is a lot of misinformation circulating that does not take into account that everyone has different levels of pain tolerance and toughness.
 But these examples debunk the notion that is it flat-out impossible for a fighter to throw punches with a torn rotator cuff.  

 

There is even a major TV network announcer (who I won't name) that is going as far as making misleading statements such as stating that when a fighter has a torn rotator cuff, it is impossible to lift his arm, which is completely false and a poor attempt to smear Pacquiao's name.
 


However, don't take my word for it, take Evander Holyfield's account where he explains what it's like fighting with a torn rotator cuff injury. For comparison purposes, I will provide excerpts from Evander and Bernard Holyfield's book "The Humble Warrior" and you can make your own conclusions and comparisons to Pacquiao's situation.
 


"Thirty minutes before the Moorer fight, Evander went through some of his routine stretches--only this time, while moving his left shoulder, he suddenly halted and grimaced. Every face in the locker room became serious. Tim Hallmark stepped forward. "What's wrong Holy?" (Holyfield, p. 205)
 
"My shoulder," Evander responded. "The shoulder had continued to bother him during training. When it got really bad, a doctor was usually able to reduce the discomfort to somewhere between irritating and bearable, but because of his training, the shoulder had never been given a chance to fully heal." (Holyfield, p. 205)
 
"Faced with such a serious injury, most fighters would have pulled out of the bout long before it got this close. Evander had had several opponents (including Tyson) cancel a bout because of a training-induced injury. But Evander is incapable of willingly quitting anything. He had always trained through bruises, aches, and pains. Rather than withdraw, Evander fought through such annoyances, so he naturally assumed that the sore shoulder wouldn't be any different." "He'd fight through this too" (Holyfield, p. 205).
 
"A sore shoulder before a fight signals serious trouble, and the Holyfield camp found itself in the position of trying to remain positive just as a tank was driving through its front door. Hallmark immediately began massaging Evander's shoulder, trying to loosen it up. Evander finally shrugged and said he would be fine." (Holyfield, p. 206)
 
"Then, during the second round, Evander connected with a strong left hook to Moorer's chin. The challenger dropped like a bowling pin. Just as Evander unleashed his shot, however, he felt an excruciating pain rip through his left shoulder. Suddenly, his upper body felt like it was on fire, and the nerve endings were screaming underneath his skin." (Holyfield, p. 206)
. 


"Evander knew he had two tasks in front of him: take out the wounded man in front of him and find a way to mask his injury. He didn't want either his face or his swings to signal that anything was wrong with his shoulder, but he had never felt a pain like this before." (Holyfield, p. 206)
 
"As the two fighters reengaged, Evander lifted his left hand and held it out in a pawing motion." "The arm had almost no strength. As far as a heavyweight was concerned, the arm was virtually useless." 
 
"I'm gonna have to fight this man with one arm," Evander said. 
 
"Evander uses his left jab to set up his combinations and hooks. As he gingerly moved his shoulder in the third round, he was unable to get it to respond. Instead of jabbing and following through with a combination, Evander began gently pawing with his left hand, using it only as a distraction, while he waited to unload with his right arm. If he could last one or two more rounds, Evander figured the shoulder might come back. It never did. Evander was forced to fight Moorer with one arm for the rest of the bout. His courage and resolve were enough to conceal most of the damage, however, and amazingly, Evander stayed in the fight right until the end." (Holyfield, p. 207)
 
After the fight, "back in the locker room, Evander told his doctor that his shoulders were killing him. The doctor immediately suspected (correctly) a torn rotator cuff injury." (Holyfield, p. 211) .

Ultimately, Holyfield ended up losing his first fight against Moorer. However, given the fact that Holyfield was able to fight with a torn rotator cuff,  debunks the notion that it is impossible to fight and throw punches with a torn rotator cuff. Furthermore, it is vitally important for everyone to understand that each individual person has their own pain tolerance levels. What's more, all of the aforementioned fighters I named, such as Berto, Klitschko, Chavez, were all able to fight with their torn rotator cuffs even though they tore their rotator cuffs early in their respective fights.
 
I urge anyone to go watch the Holyfield-Moorer I fight and observe Holyfield throwing combinations and utilizing his left hand with a torn rotator cuff. 

Similarly, watch Berto vs. Karass, Vitali Klitschko vs. Byrd, and Morales vs. Chavez and watch them utilize their arms that had rotator cuff injuries. If people are calling Pacquiao a liar, does that mean they are calling all of these fighters liars too?
 
I am not in any way, shape, or form indicating that the Mayweather-Pacquiao fight would have had a different outcome with a healthy Pacquiao. However, Pacquiao should be given the benefit of the doubt and he should just move on with his career.