Romans 8:38-39: "38 For I am sure that neither death nor life, nor angels nor rulers, nor things present nor things to come, nor powers, 39 nor height nor depth, nor anything else in all creation, will be able to separate us from the love of God in Christ Jesus our Lord."
Joshua 24:16: "Then the people answered, 'Far be it from us that we should forsake the LORD to serve other gods'"
Psalm 73:23: "Nevertheless, I am continually with you; you hold my right hand."
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The three cited passages all seem to inform the hymn more generally than specifically. The greatest similarity between these citations and the hymn text is in the title line ("From God can nothing move me").
The lines "In His good time He changes / All sorrow into joy" in the third verse bear some resemblance to Psalm 30:11: "You have turned for me my mourning into dancing; you have loosed my sackcloth and clothed me with gladness."
The line "He ev'ry gift imparts" in the fourth verse may refer to James 1:17: "Every good gift and every perfect gift is from above, coming down from the Father of lights with whom there is no variation or shadow due to change."
This is a minor point, but there's a temporal merism ("in evening and in morning") in the second half of the first verse: "He stretches out His hand / In evening and in morning, / My life with grace adorning / Wherever I may stand." The imagery here may come from Psalm 145:16: "You open your hand; you satisfy the desire of every living thing."