Nejak takto?
(neskúmal som Tvoj kód, riešim Tvoje slovné vysvetlenia):

Sub ConsCopy()

Riad = ActiveCell.Row
Den = Range("I" & Riad).Value

Dalsi:

If Range("A" & Riad) = "" Or Range("H" & Riad) <> "" Then
    MsgBox "Nekorektný riadok", vbCritical
    Exit Sub
End If

For i = 0 To 6
        Range("A" & Riad).Offset(0, i).Copy
        MsgBox "Obsah Clipboard-u :  " & Range("A" & Riad).Offset(0, i), vbInformation
Next i

Application.CutCopyMode = False
MsgBox "Koniec riadka", vbExclamation
Range("H" & Riad) = "x"

Do Until Range("I" & Riad) <> Den And Range("H" & Riad) <> "x"
    Riad = Riad + 1
Loop

Range("A" & Riad).Select

If Range("I" & Riad) = "" Then MsgBox "Dospel som na koniec databázy", vbInformation

End Sub