The First Way sees all motion as something caused and at least as initiated. Inertial motion, so it seems, need not be caused or initiated. There are many lengthy responses to this, but one shorter one that I haven’t seen might be called the unification response, since it turns on the fact that inertia makes a unification between motion and rest:
To consider something so far as it is moving inertially is to consider a motion so far as it is no different from rest.
The First Way does not consider motion so far as it is no different from rest.
As long as one admits that there is at least one coherent way of considering motion as different from rest, then the First Way can proceed just fine.
Newton would not agree that inertial motion is exactly the same as rest: he argued for an absolute space that gave an ontological reality to inertial motion, even if one could not mark it out with a metric. This absence of a metric is itself very problematic, but with the loss of absolute space, inertia comes to this.
Another difference is that Newton saw inertia as only active in the change of motion, not in the continuance of it. But this latter way is exactly how we have to see it in order for it to be an objection to the First Way.