Short answer: Yes — the charger IC here, IC2 (MCP73871), integrates reverse/backflow protection.
What the datasheet says (verbatim figures)
Reverse‑blocking protection on input: “provides protection from a faulted or shorted input,” preventing the battery from discharging into the input through internal body diodes (Section 6.1.1.4 Reverse‑Blocking Protection).
Reverse discharge limits:
UVLO: “battery reverse discharge current is less than 2 μA.”
Electrical characteristics (Output reverse leakage current, IDISCHARGE): 30–40 μA in Shutdown (various conditions); −6 to −13 μA at Charge Complete.
Power‑path with ideal‑diode operation: Low‑loss power‑path management; when input current is insufficient, the ideal‑diode allows the battery to supplement the system load (this is intended behavior, not a backflow fault).
Schematic note
The anodes of STAT1, STAT2, and PG LEDs are tied to the input net (“Net (CIN - P2)”), so the PG/STAT outputs are not pulled above VIN. The datasheet cautions that PG “must not be pulled up higher than VIN” due to an internal diode path back to VIN; your wiring is consistent with that.