I suppose it depends on how you define 'liturgy' and then whether you're following a strict or loose interpretation of the word. The Catholic Encyclopedia has the following entry for 'Liturgy' but while I don't think anyone would assert that Catholics don't have a liturgy, the following definition doesn't seem to precisely qualify what is and is not Liturgy:
Liturgy (leitourgia) is a Greek composite word meaning originally a
public duty, a service to the state undertaken by a citizen. Its
elements are leitos (from leos = laos, people) meaning public, and
ergo (obsolete in the present stem, used in future erxo, etc.), to do.
From this we have leitourgos, "a man who performs a public duty", "a
public servant", often used as equivalent to the Roman lictor; then
leitourgeo, "to do such a duty", leitourgema, its performance, and
leitourgia, the public duty itself.
So in Christian use liturgy meant the public official service of the
Church, that corresponded to the official service of the Temple in the
Old Law.
We must now distinguish two senses in which the word was and is still
commonly used. These two senses often lead to confusion.
On the one hand, liturgy often means the whole complex of official
services, all the rites, ceremonies, prayers, and sacraments of the
Church, as opposed to private devotions. In this sense we speak of the
arrangement of all these services in certain set forms (including the
canonical hours, administration of sacraments, etc.), used officially
by any local church, as the liturgy of such a church — the Liturgy of
Antioch, the Roman Liturgy, and so on. So liturgy means rite; we speak
indifferently of the Byzantine Rite or the Byzantine Liturgy. In the
same sense we distinguish the official services from others by calling
them liturgical; those services are liturgical which are contained in
any of the official books (see LITURGICAL BOOKS) of a rite. In the
Roman Church, for instance, Compline is a liturgical service, the
Rosary is not.
The other sense of the word liturgy, now the common one in all Eastern
Churches, restricts it to the chief official service only — the
Sacrifice of the Holy Eucharist, which in our rite we call the Mass.
This is now practically the only sense in which leitourgia is used in
Greek, or in its derived forms (e.g., Arabic al-liturgiah) by any
Eastern Christian. When a Greek speaks of the "Holy Liturgy" he means
only the Eucharistic Service. For the sake of clearness it is perhaps
better for us too to keep the word to this sense, at any rate in
speaking of Eastern ecclesiastical matters; for instance, not to speak
of the Byzantine canonical hours as liturgical services. Even in
Western Rites the word "official" or "canonical" will do as well as
"liturgical" in the general sense, so that we too may use Liturgy only
for the Holy Eucharist.
