The person credited with first recognizing (in the 1860s) that living cells cannot arise spontaneously, but arise only from previously existing cells, is (see book section: overview: the key roles of cell division)

The person credited with first recognizing (in the 1860s) that living cells cannot arise spontaneously, but arise only from previously existing cells, is Rudolf Virchow.
He is German physician and pathologist, who invented the modern concept of pathological processes. He used the cell theory to explain the effects of disease in the organs and tissues of the body.