My understanding would be that this "building" or expansion of His church was fulfilled in the great commission and the Spirit being sent to carry the gospel into all nations. Maybe I am not reading enough into this. I'll see what others have to say...
well it's very good to see in the correct text "I will build my church" this seems to involve Peter as well. Here the correct saying by JSauer build my church not begin my church. Real simple God established His in the beginning. I just looked up while typing this about Peter being the first Pope and they the RCC cannot really explain that very well unless they add people to the story that don't exist. btw I know we are not talking about Peter.
This may be somewhat tangential, but I have found that it's hard for modern men to appreciate the importance attached to formal changes in Scripture and instead are apt to see only material changes as significant. That is to say, the Bible seems to treat something which is new formally--i.e. in administration and/or redemptive historical context--as new indeed while we feel we must search for a more fundamental, material change to do justice to its "newness". We see this often in debates over worship, but here we can perhaps see Christ truly building a church in NT times in terms of its administrative and redemptive historical aspects while the underlying material reality of invisible and visible covenant communities remains basically the same, if that makes any sense.
I think it would be based on Peter's confession of Who Jesus is, which was revealed when Our Lord came in the flesh. I agree the church was already in existence but now that Jesus came, the church from Pentecost on will be built on the clear revelation of Who He is.
The Church from the first century on is built by the Theanthropos to whom all power in Heaven and on Earth was given after His resurrection and ascension as Divine Mediator. He will build His Church and at the Eschaton present it to His Father (I Corinthians 15).
This is something new. Jesus Christ as to His human nature, the same as who said these words to His disciples in His humiliation, is now on the throne of the Universe.
Despite the fact that the word "build" does not necessarily mean "build from scratch," I think we can acknowledge that Mat. 16:18 does include some sense of a new beginning. This is not a problem. Reformed folk recognize that the church has, in some sense, a new beginning with the post-crucifixion/resurrection/ascension era, just as we recognize that the covenant is in some sense a "new covenant." Saying so does not mean the new covenant is not also a continuation of the same covenant that has long been in place, and saying that the church has a new beginning does not mean it is not also a continuation of the church that has existed since "men began to call upon the name of the Lord" (Gen. 4:26).
Don't let anyone push you into a single, textbook definition of the term "church." Jesus is not so constrained. There is a sense in which the church becomes more fully his church following his ascension; he is exalted to rule over the church. The fact that he speaks in such terms does not mean he would disapprove of us also speaking of the church as inclusive of all his people from all time.