Objection 1. It seems that God is not everywhere. For to be everywhere means to be in every place. But to be in every place does not belong to God, to Whom it does not belong to be in place at all; for "incorporeal things," as Boethius says (De Hebdom.), "are not in a place." Therefore God is not everywhere.
Objection 2. Further, the relation of time to succession is the same as the relation of place to permanence. But one indivisible part of action or movement cannot exist in different times; therefore neither can one indivisible part in the genus of permanent things be in every place. Now the divine being is not successive but permanent. Therefore God is not in many places; and thus He is not everywhere.
Objection 3. Further, what is wholly in any one place is not in part elsewhere. But if God is in any one place He is all there; for He has no parts. No part of Him then is elsewhere; and therefore God is not everywhere.
On the contrary, It is written, "I fill heaven and earth." (Jeremiah 23:24).
I answer that, Since place is a thing, to be in place can be understood in a twofold sense; either by way of other things--i.e. as one thing is said to be in another no matter how; and thus the accidents of a place are in place; or by a way proper to place; and thus things placed are in a place. Now in both these senses, in some way God is in every place; and this is to be everywhere.
First, as He is in all things giving them being, power and operation; so He is in every place as giving it existence and locative power. Again, things placed are in place, inasmuch as they fill place; and God fills every place; not, indeed, like a body, for a body is said to fill place inasmuch as it excludes the co-presence of another body; whereas by God being in a place, others are not thereby excluded from it; indeed, by the very fact that He gives being to the things that fill every place, He Himself fills every place.
Reply to Objection 1. Incorporeal things are in place not by contact of dimensive quantity, as bodies are but by contact of power.
Reply to Objection 2. The indivisible is twofold. One is the term of the continuous; as a point in permanent things, and as a moment in succession; and this kind of the indivisible in permanent things, forasmuch as it has a determinate site, cannot be in many parts of place, or in many places; likewise the indivisible of action or movement, forasmuch as it has a determinate order in movement or action, cannot be in many parts of time. Another kind of the indivisible is outside of the whole genus of the continuous; and in this way incorporeal substances, like God, angel and soul, are called indivisible. Such a kind of indivisible does not belong to the continuous, as a part of it, but as touching it by its power; hence, according as its power can extend itself to one or to many, to a small thing, or to a great one, in this way it is in one or in many places, and in a small or large place.
Reply to Objection 3. A whole is so called with reference to its parts. Now part is twofold: viz. a part of the essence, as the form and the matter are called parts of the composite, while genus and difference are called parts of species. There is also part of quantity into which any quantity is divided. What therefore is whole in any place by totality of quantity, cannot be outside of that place, because the quantity of anything placed is commensurate to the quantity of the place; and hence there is no totality of quantity without totality of place. But totality of essence is not commensurate to the totality of place. Hence it is not necessary for that which is whole by totality of essence in a thing, not to be at all outside of it. This appears also in accidental forms which have accidental quantity; as an example, whiteness is whole in each part of the surface if we speak of its totality of essence; because according to the perfect idea of its species it is found to exist in every part of the surface. But if its totality be considered according to quantity which it has accidentally, then it is not whole in every part of the surface. On the other hand, incorporeal substances have no totality either of themselves or accidentally, except in reference to the perfect idea of their essence. Hence, as the soul is whole in every part of the body, so is God whole in all things and in each one.
Source: Summa Theologica (New Advent)