Since this is the Traditional Catholic forum, it seems worth presenting the older Catholic practice:
Communicatio in sacris
The traditional teaching of Catholic theology on whether Catholics may participate in non-Catholic religious services is summed up by St Alphonsus Liguori in his Theologia Moralis. This doctor of the church writes, ‘It is not permitted to be present at the sacred rites of infidels and heretics in such a way that you would be judged to be in communion with them’.1 …]
This teaching does not imply that the simple presence of a Catholic at a non-Catholic religious service is a sin. Thus moral theologians prior to Vatican II, following the lead of St Alphonsus, acknowledge that there may be a good reason for a Catholic to attend such a service, as when friendship leads one to attend a non-Catholic wedding. This is called by some theologians ‘passive communicatio in sacris’. It is active participation in a non-Catholic religious service which is forbidden by the traditional teaching on communicatio in sacris, for example joining in with psalms and hymns in the course of a Lutheran Eucharist. The following examples may serve to show the unanimity of pre-conciliar theologians on this point.
Fr. D. Prummer OP, writing in 1910, affirms in his Manuale Theologiæ Moralis that it is never licit for a Catholic to take part in a non-Catholic cult with the intention of worshipping God in the manner of non-Catholics, more acatholicorum. Such an act, he declares, is nothing other than a denial of the Catholic faith.2 In the same year, writing an article on ‘Heresy’ for the Catholic Encyclopœdia, Fr. J. Wilhelm SJ affirms that a Catholic may attend non-Catholic services, but only ‘provided no active part be taken in them’. In an article on the same subject, the Dictionnaire de théologie catholique reiterates, in 1920, that active participation in non-Catholic rites is toujours interdite – the reason being that it is ‘equivalent to a denial of the Catholic faith’. In 1930, Fr. B. Merkelbach OP in his Summa Theologiæ Moralis writes that ‘active participation in the sacred things of a [non-Catholic] public cult is illicit, since it implies approval of the worship and a recognition of the sect.’3 Using a slightly different terminology but teaching the same doctrine, Fr. L. Fanfani OP writes, in 1950, ‘material communicatio in sacris ‘material’ in the sense that the person in question does not mean to renounce his Catholic faith], if it is active and immediate, is never permissible for Catholics.’4 The reason for this, he explains, is that such behaviour necessarily manifests a commitment to a heretical or at least an illegitimate cultus.
It is important to notice that this prohibition is not presented by these theologians as an ecclesiastical ban. It is not the law of the Church which is traditionally understood to exclude Catholics from taking part in non-Catholic services; it is the divine law, which requires that outward acts of worship be expressive of inward faith. Nor is common worship only forbidden when the prayers or scriptural translations used by the non-Catholic group have an heretical sense: the mere act of sharing the worship of a non-Catholic group, according to the teaching of the theologians cited above, implies a community of religion with that group, and hence constitutes a sin against the faith. This explains why, as Pius XI recalled in the 1928 encyclical Mortalium Animos, ‘[the] Apostolic See has never allowed its subjects to take part in the assemblies of non-Catholics.’
I do not wish to cast aspersions on current practice or endorse the cited priest as certainly correct in his analysis. But coming up with a traditional approach to the moral question is more complicated than simply determining current discipline, and the older approach challenges us to ask whether worshipping with the Orthodox would be prudent, even if it may be allowed.