Update from January 2023.

XlsxWriter 3.0.6+ now supports a autofit() worksheet method:

from xlsxwriter.workbook import Workbook

workbook = Workbook('autofit.xlsx')
worksheet = workbook.add_worksheet()

# Write some worksheet data to demonstrate autofitting.
worksheet.write(0, 0, "Foo")
worksheet.write(1, 0, "Food")
worksheet.write(2, 0, "Foody")
worksheet.write(3, 0, "Froody")

worksheet.write(0, 1, 12345)
worksheet.write(1, 1, 12345678)
worksheet.write(2, 1, 12345)

worksheet.write(0, 2, "Some longer text")

worksheet.write(0, 3, "http://ww.google.com")
worksheet.write(1, 3, "https://github.com")

# Autofit the worksheet.
worksheet.autofit()

workbook.close()

Output:

Or using Pandas:

import pandas as pd

# Create a Pandas dataframe from some data.
df = pd.DataFrame({
    'Country':    ['China',    'India',    'United States', 'Indonesia'],
    'Population': [1404338840, 1366938189, 330267887,       269603400],
    'Rank':       [1,          2,          3,               4]})

# Order the columns if necessary.
df = df[['Rank', 'Country', 'Population']]

# Create a Pandas Excel writer using XlsxWriter as the engine.
writer = pd.ExcelWriter('pandas_autofit.xlsx', engine='xlsxwriter')
df.to_excel(writer, sheet_name='Sheet1', index=False)

# Get the xlsxwriter workbook and worksheet objects.
workbook = writer.book
worksheet = writer.sheets['Sheet1']

worksheet.autofit()

# Close the Pandas Excel writer and output the Excel file.
writer.close()

Output:

I only know of a way to do this with COM.

import contextlib, os, win32com.client

@contextlib.contextmanager
def load_xl_file(xlfilepath):
    ''' Open an existing Excel file using a context manager 
        `xlfilepath`: path to an existing Excel file '''
    xl = win32com.client.DispatchEx("Excel.Application")
    wb = xl.Workbooks.Open(xlfilepath)
    try:
        yield wb
    finally:
        wb.Close(SaveChanges=True)
        xl.Quit()
        xl = None # this actually ends the process 

def xlautofit(xlfilepath,skip_first_col=False):
    ''' relies on win32com.client to autofit columns on data sheets 

        remember that this is using COM so sheet numbers start at 1 (not 0), 
        so to avoid requiring the caller to remember this, we increment 

        returns full path (including dir) to file '''
    if os.path.splitext(xlfilepath)[1] not in ('.xls','.xlsx'):
        raise 
        return -1

    autofitbegcol = 1
    if skip_first_col:
        autofitbegcol += 1

    # Autofit every sheet 
    with load_xl_file(xlfilepath) as wb:
        for ws in wb.Sheets:
            autofitendcol = ws.UsedRange.Columns.Count
            ws.Range(ws.Cells(1, autofitbegcol), 
                     ws.Cells(1, autofitendcol)).EntireColumn.AutoFit()
    return xlfilepath