These are two topics dear to my heart. Before today, I hadn't thought much of them as being related, but a few Providential nudges hither and thither have brought their relationship into focus in my mind.
First, brief definitions and descriptions: Inculturation is the process of incarnating the gospel in a culture that previously has not received the Faith. Inculturation is analogous to translation: when someone translates, say Goethe's Faust from German into English, it becomes a slightly different work. Walter Kaufmann thought that it started to sound too serious, since the jokes in it do not translate (though this might have to do with German humor, too). On the other hand, translation can add things to the text that weren't there before and may even add to the text. Milton's translations of the Psalms turn out to be interpretations of the Psalms, not at all equivalent to the original sense, even if still faithful to it. In a similar way, inculturation requires the new culture to adopt some unfamiliar words and concepts: they simply don't translate into the new language. At the same time, the culture receiving the gospel adds to the gospel. I don't mean this in a technical sense (the deposit of the Faith is complete), but in the sense that the interpretation of the gospel is never finished, and each culture has its own inherent genius to add to the overall understanding of the gospel. Two famous examples would be: 1) the Creed's use of Greek philosophy to sort out the nature of God. The Bible guided much of the language, but so did Plato, Aristotle and Stoicism; 2) the borrowing of vestments (e.g. chasubles and miters) and other items from Roman civil religion (e.g. the word pontiff, which originally was applied to the high priest of the Roman cult).
Inculturation is tricky business and not infrequently scandalizes some who feel that the translation of the gospel into new cultures adulterates and corrupts it. Nevertheless, the Catholic Church is officially committed to it today. The recognition of the idea is relatively new: I believe that Karl Rahner coined the word in the 50's or 60's. Pope John Paul II, not surprisingly, was the first pope to use the word in official documents, including his great encyclical on missionary activity Redemptoris Missio. [BTW, is there a better website than the Vatican's?]
Ecumenism, on the other hand, is an old idea from an old word: oikos, 'house'. Ecumenism is about taking care of internal relations and workings of the Church. Today, we normally use this term when discussing the dialogue taking place between Catholics, Orthodox and Protestants.
So what is the relationship between ecumenism and inculturation? At the level that I see the two today, the important link is that in both cases the 'catholic project' (that is, the goal of uniting Christians into one Church, which typically centers in some way or other on the Catholic Church, for a variety of reasons) can appear threatened by diversity in cultural practice. The question, "Can we allow Greek philosophy into Christian theology" finds a parallel from the Catholic perspective in "Can we allow Lutheran devotions into Catholic practice?"
One of the great struggles faced by 'converts' (see below) to Catholicism is that they are often expected to abandon old forms of piety and adopt new, without much support for either. Personally, I believe that this is questionable. For example: I have spoken with several Protestants in recent years whose principal obstacle to union with Catholics is the rosary (prayer to Mary and the saints is a larger general problem). Now, the rosary is a great and powerful prayer, a great means to contemplation. However, it is far from essential to the Church. In fact, it is a kind of distillation of the much earlier and fundamental monastic practice of praying 150 Psalms each week (the original three decades replaced the Psalms with 150 Hail Marys). The rosary is highly recommended in Benedictine monasteries as a means of devotion to Mary, but is not required. Is it misplaced zeal that makes this a touchstone of Catholic unity and identity, and thus puts a stumbling block in front of potential 'converts'?
On a similar note: Anglicans have all sorts of practices that are very close to Catholic practices. My favorite example is Evensong. Catholic parishes are supposed to be promoting Vespers in these post-Vatican II days. Can you name two parishes (with diocesan clergy) that you know that are doing Vespers? Anglican converts I have known have frequently lamented the poor liturgy in Catholic parishes and the absence of liturgies like evensong. Why couldn't English-speaking Catholics especially find a way to incorporate this expression of praise to God?
Ditto for Lutherans who make use of Bach's cantatas: surely they have parallels in Catholic liturgy.
My main point is this: I realize that there are serious differences in theology among the different 'ecclesial communities' that I am citing here. In practice, however, many people are ultimately swayed to remain separated or take the plunge into unity based more on affective and cultural considerations (consider Chesterton's long-delayed move to Catholicism out of consideration for his wife or Evelyn Waugh's fascination with the cultural beauty of the old Catholic families). If Catholics require those seeking unity to be stripped of all former comfort and to adopt what seem like strange practices, well, that is frankly asking a lot of our fellow baptized. This is particularly odd when you considet that Catholics often seem to find it easier to deal with truly syncretistic additions to the Faith (Buddhist meditation, Teilhard's evolutionary near-pantheism, guitar music) than with pious practices of Protestants (reading Scripture [!%#$!], devotions, perhaps certain types of hospitality). I suspect that some of what drives this is simple bad feeling triggered by habitual peremptory judgments, not unlike the irritation we have getting along with a sibling (whom we know perfectly well) when we are willing to cut total strangers as much slack as they want.
Lastly, the Catholic Church stands to gain immensely by a potentially more positive approach to healthy Protestant culture. Those called to unity ('converts'--I don't care for the term: we all need conversion!) with the Catholic Church already bring tremendous zeal and energy. Are there other ways in which they can enrich the church and hasten full communion?