It could happen, definitely could happen and we, Catholics pray for the coming of the Kingdom every day. We also pray that Christ will return every day (hopefully not in Winter). And these are two different and similar things.
The Kingship of God is that we live with Christ as our sovereign. The Kingdom is what Jesus was preaching about while He built His Church. Fr. Richard Simon explains it that way on his radio show on relevant radio, we are to preach the kingdom and build the church and a lot of people outside the Catholic Church think we're called to build the kingdom. I think they're forgetting what Jesus said about His Kingdom not being of this world. (Jn 18:36).
The other problem with interpreting that Isaiah is saying that the Earthly Reign of God has already started is that in one sense it has. Everything that Isaiah said about Jesus has already come to pass and a lot still has to.
Finally, it should be noted that there can be no Catholic Interpretation of Sacred Scripture. You can only get a whiff of what it means. Some of what this is talking about seems like it should be considered in the eternal sense; Interpreted as what it means with regard to our final destination.
The closest I can do is give you the tiny notes the US Catholic Bishops conference wrote for that verse
Once the nations acknowledge God as sovereign, they go up to Jerusalem to settle their disputes, rather than having recourse to war
And that would suggest that this has not come to pass, since all nations have not acknowledged God as sovereign.