Paul can't be read to support a non-physical resurrection, in this passage or any other, unless you take his words out of context.
N. T. Wright is certainly the person to ask on the topic and he neatly summarizes the argument in an article addressing four reviews of his The Resurrection of the Son of God:
[Michael] Goulder, by contrast, proposes that the Jerusalem leadership held the view that Jesus’ resurrection was a matter of ‘spiritual’ transformation, rather than the ‘bodily resurrection’ which he ascribes to Paul. This is remarkable in itself; Goulder, never one to shirk controversial proposals, has stood on its head a more usual position, which is that Paul held a ‘spiritual’ view of the resurrection (based on the common misreading of the soma pneumatikon in 1 Corinthians 15) while some other, less Hellenized and more Jewish, early Christians stuck to a view of bodily resurrection.
What Wright calls "the common misreading", comes from 1st Corinthians 15:42-49 (ESV):
So is it with the resurrection of the dead. What is sown is perishable; what is raised is imperishable. It is sown in dishonor; it is raised in glory. It is sown in weakness; it is raised in power. It is sown a natural body; it is raised a spiritual body. If there is a natural body, there is also a spiritual body. Thus it is written, “The first man Adam became a living being”; the last Adam became a life-giving spirit. But it is not the spiritual that is first but the natural, and then the spiritual. The first man was from the earth, a man of dust; the second man is from heaven. As was the man of dust, so also are those who are of the dust, and as is the man of heaven, so also are those who are of heaven. Just as we have borne the image of the man of dust, we shall also bear the image of the man of heaven.
The word spiritual throughout the passage is pneumatikos <4152>, which can mean:
1) relating to the human spirit, or rational soul, as part of the man
which is akin to God and serves as his instrument or organ
1a) that which possesses the nature of the rational soul
2) belonging to a spirit, or a being higher than man but inferior to God
3) belonging to the Divine Spirit
3a) of God the Holy Spirit
3b) one who is filled with and governed by the Spirit of God
4) pertaining to the wind or breath; windy, exposed to the wind, blowing
While the idea of "spiritual renewal" seems possible to modern readers, in the context of resurrection, which meant a bodily coming back to life, it doesn't work. Paul uses the word "spiritual" because he is struggling to describe the sort of body that will be raised. In verses 35-49, he compares the process to the process of burying a seed. What is planted in the ground does not look like what eventually grows up, but both are of the same kind. In the Resurrection, we don't get an identical copy of our bodies, but something better. "Spiritual", in this case, is of the second definition: "belonging to a spirit, or a being higher than man but inferior to God". (It also might include the third meaning: "belonging to the Divine Spirit".)
Wright's book goes into great detail about what might and what might not be meant by resurrection in the New Testament. Here's a summary from another of his articles:
The first point to make here is vital. I have argued that the early Christians looked forward to a resurrection which was not a mere resuscitation, nor yet the abandonment of the body and the liberation of the soul, but a transformation, a new type of body living within a new type of world. This belief is embroidered with biblical motifs, articulated in rich theology. Yet in the gospel narratives we find a story, told from different angles of course, without such embroidering and theology—told indeed in restrained, largely unadorned prose. Yet the story is precisely of a single body neither abandoned, nor merely resuscitated, but transformed; and this, though itself totally unexpected, could give rise to exactly that developed view of which I have spoken. The Easter narratives, in other words, appear to offer an answer to why the early Christian hope and life took the form and shape they did.