From the text, I think NO.
Gen 3:22 indicates that AFTER eating from the Tree of Knowledge, they needed to be prevented from eating from the Tree of Life. Thus, the expulsion from the garden.
I don't agree. I think they did. Not eating from the Tree of Life only became an issue after eating from the Tree of Knowledge. Before then, there was no restriction in the text. The only restriction was the Tree of Knowledge.
And the LORD God said, “The man has now become like one of us, knowing good and evil. He must not be allowed to reach out his hand and take also from the tree of life and eat, and live forever.” So the LORD God banished him from the Garden of Eden to work the ground from which he had been taken
After eating of the Tree of Knowledge, they had to be banished from the Garden in order to prevent them from eating from the Tree of Life. Before then it wasn't necessary.