3M Co. agreed to buy medical-products maker Acelity Inc. for about $4.3 billion, its biggest acquisition ever, as new Chief Executive Officer Mike Roman takes a more aggressive approach to expanding the beleaguered company.

The purchase, from a group of funds advised by Apax Partners, adds bandages and surgical-wound products to one of 3M’s most profitable businesses. The company valued the transaction at $6.7 billion including Acelity’s debt, which was $2.4 billion on Dec. 31.

“Health care for us has...