No, the COVID-19 vaccines do not contain any aborted fetal cells.
However, Pfizer and Moderna did perform confirmation tests (to ensure the vaccines work) using fetal cell lines. And Johnson & Johnson uses fetal cell lines in vaccine development, confirmation and production.
But it's important to have the full context: Fetal cell lines are not the same as fetal tissue.
Fetal cell lines are cells that grow in a laboratory. They descend from cells taken from elective abortions in the 1970s and 1980s. Those individual cells from the 1970s and 1980s have since multiplied into many new cells over the past four or five decades, creating fetal cell lines. Current fetal cell lines are thousands of generations removed from the original fetal tissue. They do not contain any tissue from a fetus.