My answer is NO. One should never use a prescription medication on someone (something) for which it does not have a medical doctors approval.
It is always wrong to assume a medication for a human can double for use on dogs and other animals. Some are toxic. This noted, there are in fact cases of over-the-counter and home-remedies that humans use that can be used on a dog, but not until one knows it is in general done with good success. Don't be the test case.
I know the spirit of your post was to get information - mine is just general, but does apply. The only good solution is to get a treatment from a veterinarian - cost, but maybe no more than a dinner out.
Follow-up, I looked up the medication (sorry I'm not good on medical terms) and now understand your dog has dry eye, or that may be a symptom. My Westie has been diagnosed with dry eye and was prescribed a dog ointment, in a very small tube. The vet cost was typical, the test was a simple "wick" test and while the medication is expensive, about $35 and I use it only every other day or so, watching to see if my dog show dry eye symptoms. The vet suggested twice a day, but I think that was a worse case prescription. The one tube has lasted a couple of months and looks to be good for a couple more. If used twice a day I suppose it would last only a month or so. The vet said this is not a cure, our dog will need this the rest of his life. Hope I can get by on the lower does level I've been using, so far so good.
No, you should not use a human prescription for dogs. Quite simply, we have vastly different bodies - and some human medicines can be extremely damaging to a dog. Added to this, the strength of the human vigamox will be far too concentrated to use on a dog, and you cannot dilute it down to the correct and theraputic formulation. Finally, how do you know it is conjunctivitis and even if it is, it is more important to treat the cause of it rather than the symptoms produced by it. In short ... you need to take your dog to a good vet a) for a propper diagnisis and b) to get the appropriate treatment. Tony