Question: I am quite uncomfortable with the use of the term "spiritual values" in describing human values (such as art, education, love, etc.) from an Objectivist perspective. The primary definitions of the word "spiritual" refer to concepts such as religion, church, spirit, or soul. Are we doing an unintentional disservice to Objectivist philosophy when we describe some Objectivist values as being spiritual in nature?
[In a debate with a Theist they observed we had common ground because we both believed in spiritual things.]
Answer: Objectivists mean by "spiritual values" those values that fulfill the needs of human consciousness. The word "spirit" indeed refers in general usage to the human spirit or soul. Secondarily, it is traditionally applied to activities or institutions that are meant to address the needs or nature of the human spirit. These include, as you note, religions, churches, and the like.
Religions are associated with the spiritual due to the widespread (and false) dichotomy between mind and body. If one believes that the human soul or spirit exists apart from the material and natural body, then one might well believe, too, that the "spiritual" realm refers to a supernatural dimension, such as a heaven, or to some supernatural experience, such as an encounter with God or God's messengers.
But the mind-body dichotomy is false. Human beings are unified organisms whose minds depend radically on their material bodies to exist, and whose bodies require a healthy mind to continue to live. The words "spirit" and "spiritual" refer to real aspects of human experience, namely the mental aspects of human life.
An encounter with art that calls out our emotions and sense of life, but that does not primarily address any physical need, is a spiritual encounter, and Objectivists are not the only people to see this. You can learn more about art as a spiritual experience from Alexandra York's talk on that subject from our 2004 Summer Seminar.
And so it is for the other examples you mention. The fact that there is no supernatural realm does not mean that human life is dull or empty; it does not mean that the real yearnings and idealism associated with spiritualism have no place in a rational life. This is why Ayn Rand featured a "temple of the human spirit" in her great novel, The Fountainhead. This was a temple to make man feel what was possible to him, and what greatness he embodied. To be able to project ourselves into world-spanning, destiny-altering projects is a capacity uniquely arising from our rational faculty, i.e., from our mind. To honor that capacity in man is thus to honor the human spirit.