How can I make running a command depend on the input of a correct password (Tkinter GUI)?

On the edge of being off-topic, but for the sake of gksudo:

Not sure what you want to achieve, since the GUI does not have any effect on what happens in the terminal :)

Then technically

The problem is that you do not set a condition for tkMessageBox.showinfo("Login successful!") to be executed, so whatever happens in os.system("gksudo su && sudo su"), the next line will be performed.

def root_login():
    os.system("gksudo su && sudo su")   
    tkMessageBox.showinfo("Login successful!")

How to make it work

First, you shouldn't use os.system any more: Really, really old fashioned.

See below for an alternative coding, using subprocess.check_call:

#!/usr/bin/env python
import subprocess
import Tkinter
import tkMessageBox

top = Tkinter.Tk()
top.geometry("400x250")

def root_login():
    try:
        subprocess.check_call(["gksudo", "su"])
    except subprocess.CalledProcessError:
        tkMessageBox.showinfo("message", "OOOOPS...\nWrong password!")
    else:
        tkMessageBox.showinfo("message", "Login successful!")

def close_window():
    top.destroy()

B = Tkinter.Button(top, text ="Login", command = root_login)
B.pack()
Q = Tkinter.Button(top, text ="Quit", command = close_window)
Q.pack()

top.mainloop()

But again, gksudo su does not have any effect, since you run a GUI :)

Explanation

subprocess.check_call(["gksudo", "su"])

...will raise a subprocess.CalledProcessError in case the password is incorrect, and show the message:

If the password is correct, the message:

will appear.

Note

The try/except/else construction, I did for clarity reasons. The function below does exactly the same, since the function wil "jump" from the line:

subprocess.check_call(["gksudo", "su"])

to

tkMessageBox.showinfo("message", "OOOOPS...\nWrong password!")

in case, and immediately if, the first one raises the subprocess.CalledProcessError:

def root_login():
    try:
        subprocess.check_call(["gksudo", "su"])
        tkMessageBox.showinfo("message", "Login successful!")
    except subprocess.CalledProcessError:
        tkMessageBox.showinfo("message", "OOOOPS...\nWrong password!")