Removing the existing plastic from the environment (and mitigating the damage from future releases) is a significantly greater challenge. Even if we managed to clean up the gyres in some way, degraded plastic appears to be a ubiquitous presence in the water and soil. Supposing that we managed to clean up all that was visible we would still be leaving a substantial fraction behind in the form of highly-degraded particles. It is unlikely that we will ever be able to clean it all up using conventional means. What we really need to do is to enable some creature or set of creatures to digest the stuff.
This would not be a small challenge: even natural polymers are notoriously hard for most creatures to digest. However, as I commented in the Orgel post, we know of bacteria that can digest petrochemical waste products. Given sufficiently masticated plastics, it should in principle be possible, perhaps through directed evolution, to develop proteins appropriate to degrade the polymers and digest the subunits thus produced. The natural solution to the degradation of cellulose as a food source involves perhaps as many as 200 species of bacteria and protozoans; it is unlikely we could produce anything nearly as efficient with a single microbe. However, it may be sufficient to produce just a rudimentary digestive reaction. Once that is good enough for plastic digestion to get started, efficiency is likely to evolve on its own, since the ability to utilize such a food source could constitute a significant selective advantage. Obviously one would want to use an obligate anaerobe or perhaps some kind of plankton for this purpose, to diminish the chances of a “gray goo” problem.
Given a few million years or so, it’s likely that microorganisms will evolve that can digest plastic, or help some other organism digest it (as in the case of termites). Regrettably, we do not have millions of years to solve the problem. Of course we should try to come up with some mechanical means of harvesting the larger chunks of plastic from the ocean, but in the absence of significant penalties for dumping, on an international level, such steps will only provide temporary amelioration. Moreover, they will constitute only a partial solution to the problem. In this case, it may be wise for us to try and give evolution a nudge in whatever way we can, lest the prediction of a ‘plasticene’ geological stratum come true.