How does a Catholic know that they are not worshiping the same God as someone who practices Islam or Judaism? Is there anything that concretely points to them being different past personal revelation? Has the Church expressed in a doctrine the differences between their God and the others?
Some people like to say that because the gods of Judaism, Christianity and Islam share a heritage then they're the same God. Other people point to the big doctrinal differences between them to say they're not the same. I have to go with the second group. It's kind of like the difference between The Republic of China and The People's Republic of China: both claim to be the legitimate government of China and both have ties to the historical government of China, but no one would pretend that they are united.
The three religions have these things in common:
- Their God is monotheistic
- They see Abraham as their founder
- They share the Old Testament prophets
- They believe in one life followed by judgment
Factors like these lead people to say that they are all about the one God. But factors like these, which are far more significant in my opinion, lead most people to say they are different gods:
- The Islamic god is unknowable, but the gods of Judaism and Christianity wants to be known as much as is possible
- The Islamic god loves only the righteous, but the gods of Judaism and Christianity loves the unrighteous as well
- The God of Christianity is the Trinity, which Islam and Judaism reject
- Jesus is God in Christianity, which Islam and Judiasm reject
The Catholic ought to start with
3. The Church regards with esteem also the Moslems. They adore the one God, living and subsisting in Himself; merciful and all- powerful, the Creator of heaven and earth,5 who has spoken to men; they take pains to submit wholeheartedly to even His inscrutable decrees, just as Abraham, with whom the faith of Islam takes pleasure in linking itself, submitted to God. Though they do not acknowledge Jesus as God, they revere Him as a prophet. They also honor Mary, His virgin Mother; at times they even call on her with devotion. In addition, they await the day of judgment when God will render their deserts to all those who have been raised up from the dead. Finally, they value the moral life and worship God especially through prayer, almsgiving and fasting.
5. Cf St. Gregory VII, letter XXI to Anzir (Nacir), King of Mauritania (Pl. 148, col. 450f.)
4. As the sacred synod searches into the mystery of the Church, it remembers the bond that spiritually ties the people of the New Covenant to Abraham's stock.
Thus the Church of Christ acknowledges that, according to God's saving design, the beginnings of her faith and her election are found already among the Patriarchs, Moses and the prophets. She professes that all who believe in Christ-Abraham's sons according to faith 6 -are included in the same Patriarch's call, and likewise that the salvation of the Church is mysteriously foreshadowed by the chosen people's exodus from the land of bondage. The Church, therefore, cannot forget that she received the revelation of the Old Testament through the people with whom God in His inexpressible mercy concluded the Ancient Covenant. Nor can she forget that she draws sustenance from the root of that well-cultivated olive tree onto which have been grafted the wild shoots, the Gentiles.7 Indeed, the Church believes that by His cross Christ, Our Peace, reconciled Jews and Gentiles. making both one in Himself.8
The Church keeps ever in mind the words of the Apostle about his kinsmen: "theirs is the sonship and the glory and the covenants and the law and the worship and the promises; theirs are the fathers and from them is the Christ according to the flesh" (Rom. 9:4-5), the Son of the Virgin Mary. She also recalls that the Apostles, the Church's main-stay and pillars, as well as most of the early disciples who proclaimed Christ's Gospel to the world, sprang from the Jewish people.
6. Cf. Gal. 3:7
7. Cf. Rom. 11:17-24
8. Cf. Eph. 2:14-16
The God whom the Church, the new People of God, believes in, they owe it to the revelation of the Old Testament which the Church received from the Jews, God's own chosen people.
The Catholic may then wish to move on to "Muhammad?" and "Judaism?" sections in Crossing the Threshold of Hope by Pope St. John Paul II [the Great].
The Catholic should then conclude that from these papal teachings, the Catholic Church acknowledges that Jews and Muslims worship the same God. (OP: Even though Jews have a incomplete knowledge of him and Muslims have an incomplete and distorted knowledge of him).
The context in which these declarations should be viewed
- Worshiping the same God does not imply that the various worshipers are worshiping God as he ought to be worshiped. cf. Jesus and the Woman of Samaria.
- Among those who worship God, some worship is acceptable to him and some is not. cf. Cain & Abel and Peter and Cornelius. In the case of Cornelius, even though God accepted him, God via Peter brought him to the fullness of the Faith.
- (OP: This declaration is not an endorsement of the Jews' and Muslims' worship, much less of their religion).
The Bible states that we are to "believe" in the Lord in order to have salvation. It is therefore extremely important what God says about himself sot that we can believe it. If we believe in something other than what God said he was, then we do not actually believe in Him. This is why the doctrine of the Trinity is so very important. As A.W. Tozer puts it "Among the sins to which the human heart is prone, hardly any other is more hateful to God than idolatry, for idolatry is at bottom a libel on His character. The idolatrous heart assumes that God is other than He is - in itself a monstrous sin - and substitutes for the true God one made after its own likeness." (Knowledge of the Holy, Pg. 5)
Therefore if we pre-suppose that the doctrine of the Trinity is a correct representation of what God said about himself, then any religion which does not acknowledge the Trinity and Deuteronomy 6:4 and which does not believe Isiah 44:6 which states "I am the first and I am the last; besides me there is no god" and does not believe that Jesus came and died to save us from our sins does not believe in the same God but an idol fashioned after that religions own likeness.
The traditional Catholic Church says that Muslims worship devils not the true God.
For all the gods of the Gentiles are devils: Psa 95:5
Pope Callixtus III: “I vow to… exalt the true Faith, and to extirpate the diabolical sect of the reprobate and faithless Mahomet [Islam] in the East.”
Pope Eugene IV, Council of Basel, Session 19, Sept. 7, 1434: “Moreover, we trust that with God’s help another benefit will accrue to the Christian commonwealth; because from this union, once it is established, there is hope that very many from the abominable sect of Mahomet will be converted to the Catholic faith.”