Assuming that I understand what you mean, you do not. To open your project outside of Twine 2 in a playable form, in macOS or any other, you must publish it to file and then open that compiled HTML file in your browser of choice. Check your project's story menu for the Publish to File item.
Even if using the browser-based version of Twine, as opposed to the executable version, the URL shown within the application using either of the Play or Test modes—both of which are testing modes, actually, despite the naming—is only valid within that specific instance of the application and browser, you cannot access it externally.
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Even if using the browser-based version of Twine, as opposed to the executable version, the URL shown within the application using either of the Play or Test modes—both of which are testing modes, actually, despite the naming—is only valid within that specific instance of the application and browser, you cannot access it externally.