1 Timothy 3:15 But if I tarry long, that thou mayest know how thou oughtest to behave thyself in the household of God, which is the assembly of the living God, the pillar and ground of the truth. It is the assembly of God that is to uphold and support the truth of God here on earth. That "household" consists of all true and faithful followers and believers in Christ. One of the "truths" contained in scripture is that anyone who aspires to leadership in that household is to conform to the ( ) set by the Holy Spirit through Paul.
1 Timothy 3:2 A bishop then must be blameless, the husband of one wife, vigilant, sober, of good behaviour, given to hospitality, apt to teach; 3 Not given to wine, no striker, not greedy of filthy lucre; but patient, not a brawler, not covetous; 4 One that ruleth well his own house, having his children in subjection with all gravity;
The Catholic Church couldn't even uphold that basic truth. The Catholic Church doesn't qualify.
I have to side with Mark17 on this question. The wording is not cut and dry, as the phrase in question could be describing “the church of the living God”, or simply the directly preceding “the living God”.
So I think this is a case where one needs to look to the rest of Scripture to see which is the more likely interpretation. It is God, not the church, that is consistently described in the rest of Scripture with symbols like “the rock”, “the chief cornerstone”, “the sure foundation”, etc. So, I think it is more in line with the rest of Scripture to read this description as meant to apply to God, not the church.