A little "background story":
You want to let the user choose and upload a file from your popup. But in OSX, as soon as the file-chooser dialog opens, the popup loses focus and closes, causing its JS context to get destroyed as well. Thus, the dialog opens and closes immediately.
This is a known bug on MAC for quite some time.
The solution:
You can move the dialog opening logic to the background-page, which is not affected by loss of focus. From the popup, you can send a message to the background-page, requesting to initiate the browse-and-upload process (see sample code below).
manifest.json
{
    ...
    "background": {
        "persistent": false,
        "scripts": ["background.js"]
    },
    "browser_action": {
        "default_title": "Test Extension",
//        "default_icon": {
//            "19": "img/icon19.png",
//            "38": "img/icon38.png"
//        },
        "default_popup": "popup.html"
    },
    "permissions": [
        "https://www.example.com/uploads"
        // The above permission is needed for cross-domain XHR
    ]
}
popup.html
    ...
    <script src="popup.js"></script>
</head>
<body>
    <input type="button" id="button" value="Browse and Upload" />
    ...
popup.js
document.addEventListener('DOMContentLoaded', function () {
    document.getElementById('button').addEventListener('click', function () {
        chrome.runtime.sendMessage({ action: 'browseAndUpload' });
        window.close();
    });
});
background.js
var uploadUrl = 'https://www.example.com/uploads';
/* Creates an `input[type="file]` */
var fileChooser = document.createElement('input');
fileChooser.type = 'file';
fileChooser.addEventListener('change', function () {
    var file = fileChooser.files[0];
    var formData = new FormData();
    formData.append(file.name, file);
    var xhr = new XMLHttpRequest();
    xhr.open('POST', uploadUrl, true);
    xhr.addEventListener('readystatechange', function (evt) {
        console.log('ReadyState: ' + xhr.readyState,
                    'Status: ' + xhr.status);
    });
    xhr.send(formData);
    form.reset();   // <-- Resets the input so we do get a `change` event,
                    //     even if the user chooses the same file
});
/* Wrap it in a form for resetting */
var form = document.createElement('form');
form.appendChild(fileChooser);
/* Listen for messages from popup */
chrome.runtime.onMessage.addListener(function (msg) {
    if (msg.action === 'browseAndUpload') {
        fileChooser.click();
    }
});
Heads up:
As a security precaution, Chrome will execute fileChooser.click() only if it is a result of user interaction.
In the above example, the user clicks the button in the popup, which sends a message to the background-page, which calls fileChooser.click();. If you try to call it programmatically it won't work. (E.g. calling it on document load won't have any effect.)