Is it possible to send recurring bitcoin payments via the bitcoin protocol? Is there a service which provides an API to do so?
There is no ready-made recurring payment solution at the moment, however this can be achieved at application level by constructing pre-signed transactions or using CLTV's as @Wapac suggested. CLTV approach may have a downside though, you will be paying transaction fees twice in case you decide to cancel.
A more sophisticated mechanism may be creating a non-custodial, multi-sig wallet service that keeps track of your recurring payments and notify you if any action is needed. Disclosure, I am involved in such a solution called: https://www.arf.one
What you can do with Bitcoin protocol is to time-lock your transaction - see BIP 65. This allows you create a transaction that the recipient can not spend for a certain number of blocks while you can (or vice versa, which is more common). So imagine that you create and publish that transaction and set that the recipient could not spend it for about 1 month. This is a building element you can use to create something like recurring payments. For a year long month-by-month subscription (but cancel anytime) you would need 12 of those time-locks. You would publish it and once the recipient would see the transaction confirmed, they would know "this guy just created monthly subscription with us and unless he cancels, we will be able to get the money each month for next 12 months". Obviously, if you create 12 of those and you'd like to have longer subscription, you would need to create more of those later to "prolong". All this can be implemented in the user wallet including some kind of notification that your subscription needs to be prolonged and if you want to do this etc. That's more about UI and UX, however.
So in general, I'd say yes. On API side - depends, there are libraries for various languages for creating your custom transactions, but this is probably not what you were asking for as I believe you were looking for something higher level than manual transaction building.