No, it isn't. Plastic is only showing up in large quantities in areas where it shouldn't, and occasionally hurting some animals. Microplastics in particular may become a problem at some point, but then again, maybe they won't. We aren't sure. And either way, it's not causing problems just yet.
Greenhouse emissions are what's killing the planet. Manufacturing plastics is a part of that, but so is power generation, and thus all "purely digital" activity.
It's a point that people often confuse. Microplastics are bad, and emissions are bad, but they're bad for completely different reasons. Climate is what's an existential threat to civilization, and what we need to focus on right now. We can deal with microplastics later, if we survive that long.