After persistent rain had delayed the start of the contest, the umpires deemed the conditions safe enough for a reduced nine overs per side match.
Having lost the toss, South Africa batted first and posted 77-5, with Hashim Amla again stealing the show with an unbeaten 47 from just 30 balls.
With England needing to reach 41 by the end of the fifth over without losing any further wickets, to be ahead on the Duckworth-Lewis system, Wright was caught in the deep by Morne Morkel.
Wright's dismissal left South Africa firmly in control, but before play could resume, the rain reappeared, leading to a no result.
England will be looking to level the three-match series when they face the Proteas in the final match at Edgbaston on Wednesday.