According to three different Greek Church Fathers, Paul is referring to the Holy Spirit here.
Cyril of Jerusalem refers to this verse in his 17th Catechetical Lecture ("Continuation of the Discourse on the Holy Ghost"):
Thou wilt find many other titles of the Holy Ghost besides. Thus He
is called the Spirit of Holiness, as it is written, According to the
Spirit of Holiness1
Basil the Great also refers to the verse in his discourse, On the Spirit:
You expect to be glorified together with Christ; (“if so be that we
suffer with him that we may be also glorified together;” [Romans
8:17]) but you do not glorify the “Spirit of holiness” together with
Christ, as though He were not worthy to receive equal honour even with
you. You hope to “reign with” [2 Timothy 2:12] Christ; but you “do
despite unto the Spirit of grace” [Hebrews 10:29] by assigning Him the
rank of a slave and a subordinate.2
He also paraphrases Romans 1:4 in referring to the Holy Spirit as the "fount of holiness" in one of his Epistles.3
Finally, a later Greek writer, John of Damascus, also interprets the verse in this way, writing:
[Other sayings] again are for the purpose of revealing Him to us and
strengthening our faith, as, And now, O Father, glorify Thou Me with
the glory which I had with Thee, before the world was [John 17:5]. For
He Himself was glorified and is glorified, but His glory was not
manifested nor confirmed to us. Also that which the apostle said,
Declared to be the Son of God with power, according to the spirit of
holiness, by the resurrection from the dead [Romans 1:4]. For by the
miracles and the resurrection and the coming of the Holy Spirit it was
manifested and confirmed to the world that He is the Son of
God.4
1 Lecture XVII.5
2 Chapter XXVIII, "That our opponents refuse to concede in the case of the Spirit the terms which Scripture uses in the case of men, as reigning together with Christ", Article 70
3 Epistle VIII.10
4 Exact Exposition of the Orthodox Faith, IV.XVIII