Try running Excel in safe mode. Hold down the Ctrl key while opening Excel. At the prompt click Yes.
This loads Excel without any macros (likely the source of your problem) or the default document template (which may be corrupt). See if it opens and closes cleanly. Repeat. See if it opens cleanly in normal mode and closes cleanly.
The safe mode opening/closing should have replaced a corrupt normal.xlsx (default template file). If not, search for and delete normal.xlsx and normal.xls.