I think the only workaround for this would be to make a macro which:

Unprotects the sheet
Runs the spellcheck
Protects the sheet.

This macro could be assigned to the existing button or a new one. You would
need a macro to run on opening the workbook to assign it and another which
ran on closing the workbook to unassign it.

I think there may be an issue that if there is an error the sheet  may be
accessible to your users  in an unprotected state. i.e. the spellcheck will
stop for the correction to be made  and the user could then do something
else.

Thank you for the information. At this time, I think I am willing to deal
with the misspelling rather than to risk their having access to an
unprotected sheet. Thank you for adding the information that it could
happen as it checked. I appreciate it.

I had the same problem  with a Word protected  document. Here are some
pieces of code that I used to solve the problem. You should be able
to modify for an Excel spreadsheet. CheckSpelling was attached to a
toolbar button. SavedText looked for the formfields in the Word
document that my users  could actually change, CheckPassword unlocked
and relocked the document while DoFindReplace actually allowed them
to correct any misspellings.

One thing I noticed I did not add  was an error handler to this code.
What I would do was add an on error handler to use CheckPassword to
relock the spreadsheet.

Hope this helps.

Sub CheckSpelling()

Application.ScreenUpdating = False

iCount = 0
'open locked document
Call CheckPassword

'build array of all text  fields and their result
Call SavedText(wdFieldFormTextInput)

'change all the hyphens
With ActiveDocument.Content.Find
.ClearFormatting
With .Replacement
.ClearFormatting
End With
.Execute FindText:="'", ReplaceWith:="'", Format:=True, _
Replace:=wdReplaceAll
End With

'change all the quotes
With ActiveDocument.Content.Find
.ClearFormatting
With .Replacement
.ClearFormatting
End With
.Execute FindText:=""", ReplaceWith:=""", Format:=True, _
Replace:=wdReplaceAll
End With

Application.ScreenUpdating = True

ActiveDocument.CheckSpelling

doFindReplace iCount, fField, fFieldText()

End Sub

Public Function SavedText(Optional FieldType As Integer) As String()
For Each fField In ActiveDocument.FormFields
If fField.Type = FieldType Then
ReDim Preserve fFieldText(1, iCount)

fFieldText(0, iCount) = fField.Result

fFieldText(1, iCount) = fField.Name
fField.Select

Selection.TypeText "<" & fFieldText(0, iCount) & ">"

iCount = iCount + 1

End If
Next fField

SavedText = fFieldText
End Function

Public Sub CheckPassword()
If ActiveDocument.ProtectionType = wdAllowOnlyFormFields Then
ActiveDocument.Unprotect "password"
Else
ActiveDocument.Protect wdAllowOnlyFormFields, True, "password"
End If
End Sub

Public Sub doFindReplace(iCount As Integer, fField As FormField,
fFieldText() As String)
Dim i As Integer, intRetVal As Integer
Dim strNewText As String

Selection.HomeKey Unit:=wdStory
Application.ScreenUpdating = False

For i = 0 To iCount - 1
Selection.Find.ClearFormatting
With Selection.Find
.Text = "<"
.Forward = True
.Wrap = wdFindContinue
.MatchCase = False
.Execute
End With

Selection.Extend
Selection.Extend Character:=">"

intRetVal = StrComp(Selection.Text, fFieldText(0, i))

Select Case intRetVal
Case 0
Set fField = Selection.FormFields.Add _
(Range:=Selection.Range,
Type:=wdFieldFormTextInput)
fField.Result = fFieldText(0, i)
fField.Name = fFieldText(1, i)

Case Else
intLength = Len(Selection.Text)
strNewText = Mid(Selection.Text, 2, intLength - 2)
Set fField = Selection.FormFields.Add _
(Range:=Selection.Range,
Type:=wdFieldFormTextInput)
fField.Result = strNewText
fField.Name = fFieldText(1, i)
End Select

Next i

Call CheckPassword

Application.ScreenUpdating = True

MsgBox "Spell Check Complete.", vbOKOnly, TITLE

End Sub