Naproxen is the heart-safe one. If you have any cardiovascular risk at all, that is the anti-inflammatory to reach for.
The randomised evidence is genuinely favourable on the heart. Pooled individual participant data from 280 placebo-controlled trials found no significant increase in major vascular events for high-dose naproxen (rate ratio 0.93), while coxibs and diclofenac rose by about a third (Coxib and traditional NSAID Trialists' Collaboration, 2013). Community observational data agree, placing naproxen lowest of the widely used drugs at 1.09 and risk-neutral at every dose studied (McGettigan and Henry, 2011). Two findings complicate it. In that same randomised pooling, naproxen carried the highest gastrointestinal risk in the set, raising serious upper gastrointestinal complications more than fourfold. And an observational analysis of everyday prescribing put naproxen's heart attack odds in the first week of use at 1.53, among the highest of the drugs examined rather than the lowest (Bally et al., 2017).
Naproxen trades a lower heart signal for a higher stomach one, so whether it is the safer choice depends entirely on which organ is being asked about.