Gandalf was indeed a steward of Middle-earth
The mission the Istari were sent on was to guide the free people's of Middle-earth against Sauron. As such, the five members of the order can be considered stewards of Middle-earth, however where 4 failed Gandalf succeeded. Gandalf states this quite clearly before saying the line you quote in your question.
The rule of no realm is mine, neither of Gondor nor any other, great or small. But all worthy things that are in peril as the world now stands, these are my care. And for my part, I shall not wholly fail of my task, though Gondor should perish, if anything passes through this night that can still grow fair or bear fruit and flower again in days to come. For I also am a steward. Did you not know?
Return of the King - Book V, Chapter 1: Minas Tirith
It is clear from not only this passage but also from the Istari's mission that he does indeed protect and care for Middle-earth.
In his essay titled The Istari, in the Unfinished Tales, Tolkien gives us a little insight into their mission and why it was that Gandalf was the only one to stay true to it:
for he dwelt in no place, and gathered to himself neither wealth nor followers, but ever went to and fro in the Westlands from Gondor to Angmar, and from Lindon to Lorien, befriending all folk in times of need.
This idea is further re-iterated in The Silmarillion:
But afterwards it was said among the Elves that they were messengers sent by the Lords of the West to contest the power of Sauron, if he should arise again, and to move Elves and Men and all living things of good will to valiant deeds.
Of the Ring of Power and the Third Age.
And again in Appendix B to the Lord of the Rings:
It was afterwards said that they came out of the Far West and were messengers sent to contest the power of Sauron, and to unite all those who had the will to resist him; but they were forbidden to match his power with power, or to seek to dominate Elves or Men by force and fear.
Appendix B: The Tale of Years