Here's what I wrote on another thread on Tiger.
To answer the OP, no. Running would probably not have helped prevent Tiger's back problems. Tiger Woods problems began with a pinched nerve in his back in 2013 that got progressively worse. Back problems are common among golfers due to the violence and twisting motion of the swing. He had the first of many failed back surgeries in March 2014 and two more in 2015. At this point, he has had something like 7 back surgeries!
I don't have any firsthand information on Woods' situation, but I do have firsthand experience with extreme back pain and, separately, firsthand experience with prescription opioids. Over time, tolerance to pain killers increase so the dosage has to be increased. I knew that when I took prescription opioids after a surgery, I'd need to get off them as soon as possible. What I didn't know was how incredible the withdrawal pain would be after stopping the opioid, even though I'd only taken opioids for a few days. It's easy to see how someone could get addicted to prescription opioids, especially if they were in continuous pain, as often happens after back surgery.
Back surgeries fail a lot. One back surgeon I saw with 30 years of experience told me that almost his entire practice was fixing failed back surgeries!