Let's tackle the rules first. Disciple of life establishes 3 conditions:
- use a spell
- 1st level or higher
- restore hit points to a creature
Healing spirit is a spell of 1st level or higher that you can why (Use for restore hit points. Almost word for word meets the conditions required for the Disciple of Life. This is how this feature works.
Address some concerns
It should not work because no action is required to restore hit points.
Disciple of Life does not require you to use an action, it does not mention the economy of action at all.
Also, as @jgn pointed out, it's helpful to note that Disciple of Life doesn't trigger when you cast a spell, but every time you use it to restore hit points. (This is somewhat unusual since many features are activated by casting a spell, but it is an important distinction.)
It is not the character but the spirit that is causing the healing.
The spirit is not a creature, it is not known as a creature or anything that has any mechanical impact on the game or agency other than the healing and movement you make it do. It's how the spell works. You are not commanding a spirit to do something, you are causing a physical manifestation of a spell to do it. If the spell were to summon an elemental creature that could heal on its turn, this argument would make sense, but that's not what the healing spirit does.
A similar argument could be made for the goodberry spell (he is not restoring life points, if he eats the berry), but it was clarified in the Compendium of wise advice (containing official rules on how to interpret the rules) that Disciple of life, in fact, works with the goodberry spell out:
If I am a cleric / druid with the Disciple of Life feature, does the goodberry spells benefit of function? Yes. The Disciple of Life feature would have each berry restore 4 hit points, instead of 1, assuming you cast goodberry with a first-rate spell slot.
Healing spirit it's very strong
In the end, the DM's job is to manage the game. If they feel that the healing spirit is too powerful, they can ban it or change it to make it less powerful.
If this is the case, I would suggest changing the text of the spell to:
Once per round Until the spell ends, each time you or a creature you can see moves into spirit space for the first time in a turn or starts its turn there, you can have the spirit restore 1d6 hit points to that creature (don't requires action).
(highlighted text added by me)