Yes, its possible to learn and speak druidic without being a druid.
The Faiths & Philosophies player companion introduced the Druidic Decoder feat, that allows a character to decode and even learn druidic without taking levels of druid.
This means that the language isn't magically protected from means to learn and speak it by non-druids, they merely don't share it's secrets around, nor publish any dictionaries that can be found on the local library.
Keep in mind that you do not choose languages when you cast Tongues, you simply grant the ability to speak any language of any intelligent creatures. The spell doesn't grant the knowledge of that language, it simply makes communication possible. Similarly, the spell Comprehend Languages will allow you to understand what is written on a druidic text, but not give you the knowledge to be able to translate it to common.